Formal concept analysis and lattice-valued Chu systems

Available online at www.sciencedirect.com

Fuzzy Sets and Systems 216 (2013) 52 – 90 www.elsevier.com/locate/fss

Formal concept analysis and lattice-valued Chu systems Jeffrey T. Dennistona , Austin Meltona,b , Stephen E. Rodabaughc,∗ a Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA b Department of Computer Science, Kent State University, Kent, OH 44242, USA c College of Science, Technology, Engineering, Mathematics (STEM), Youngstown State University, Youngstown, OH 44555-3347, USA

Available online 12 September 2012

Abstract This paper links formal concept analysis (FCA) both to order-theoretic developments in the theory of Galois connections and to Chu spaces or systems viewed as a common rubric for both topological systems and systems arising from predicate transformers in programming semantics [13]. These links are constructed for each of traditional FCA and L-FCA, where L is a commutative residuated semiquantale. Surprising and important consequences include relationships between formal (L-)contexts and (L-)topological systems within the category of (L-)Chu systems, relationships justifying the categorical study of formal (L-)contexts and linking such study to (L-)Chu systems. Applications and potential applications are primary motivations, including several example classes of formal (L-)contexts induced from data mining notions. Throughout, categorical frameworks are given for FCA and lattice-valued FCA in which morphisms preserve the Birkhoff operators on which all the structures of FCA and lattice-valued FCA rest; and, further, the results of this paper show that, under very general conditions, these categorical frameworks are both sufficient and necessary for the “interchange” or “preservation” of (L-)concepts and (L-)protoconcepts, structures centfral to FCA and lattice-valued FCA. © 2012 Elsevier B.V. All rights reserved. Keywords: Formal (L-)context; Galois connection; Formal (L-)concept/(L-)preconcept/(L-)protoconcept; (L-)Chu system; (L-)Topological system; (L-)formal context interchange

1. Introduction and motivations The history of mathematics seems fraught with essentially simultaneous introductions and developments of parallel ideas which were ultimately closely related but which were initiated in complete unawareness and independence of each other, independent even with regard to their respective motivations. This paper is related to such a cluster of independent and parallel ideas, all of which began in and around the 1980s. Formal concept analysis (FCA) seems to have first been introduced by R. Wille [51] in 1982, a notion which has received considerable attention due to its wide applicability and widespread mathematical connections—see the compilation in [24]. While FCA is critically tied to the notion of a Galois connection (see Section 2 below), the ordertheoretic and category-theoretic developments of Galois connections in the middle 1980s by M. Erné, J. Koslowski, J.M. McDill, A. Melton, D.A. Schmidt, G.E. Strecker [17,35,36] seem unrecognized in FCA, even when generalized to the fuzzy or lattice-valued setting. ∗ Corresponding author. Tel.: +1 330 941 3609; fax: +1 330 941 1567.

E-mail addresses: [email protected] (J.T. Denniston), [email protected] (A. Melton), [email protected], [email protected] (S.E. Rodabaugh). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.09.002

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

53

Based on preliminary notions of J.A. Goguen [26], S.E. Rodabaugh in the early 1980s proposed variable-basis topology [39,40], a notion of topology in which different spaces in the same category have different underlying carrier sets and lattices of truth values, and in which a morphism respects the change of both carrier set and membership lattice as well as the topology based upon them. Variable-basis topology was further developed by P. Eklund, Rodabaugh, S.A. Solovyov, and others—see an early and detailed history in Section 1 of [41] and compare with Section 10 of [31]. In the latter 1980s, S.J. Vickers [53] proposed the idea of topological systems, a notion capturing certain topological aspects of programming semantics. Even though Vickers proposed this notion primarily for pedagogical reasons, there has been a recent surge of interest in this idea and how it links to other mathematical areas. Indeed, the notions of variable-basis topology and topological systems are very differently motivated and originally were completely unaware of each other. But in 2009, J.T. Denniston and Rodabaugh [14] showed that variable-basis topology and topological systems are deeply related, and this is spurring further and intensive work by Denniston, C. Guido, Melton, Rodabaugh, Solovyov, and others—see [11–13,27,43,46,47] and the references of these papers. In his seminal work [15], E. Dijkstra proposed in 1976 the notion of predicate transformers as part of a strategy to improve the quality of programming. Subsequently, in 1979, Chu spaces (or Chu systems) were first introduced by M. Barr [2] as a common framework for different kinds of mathematical structures; and a survey of this field can be found in V.R. Pratt [38]. It was realized by Denniston, Melton, Rodabaugh in 2010 [11,13] that Chu systems could serve as a common rubric for both topological systems and the non-topological systems induced by predicate transformers. It is the purpose of this paper to begin tying several of the above threads together, and to do so consistent with these two boundary conditions: the level of generality should be justified by applications and potential applications; and categorical morphisms should be defined so that important structures of objects are “preserved”. Specifically, this paper does the following: • It ties FCA explicitly to the order-theoretic and category-theoretic aspects of antitone Galois connections in two ways: this paper in Section 2 uses antitone Galois connection order-theoretics to understand the equivalence classes set up by the Birkhoff operators of a formal context and how these equivalence classes relate to the notions of formal concept, formal preconcept, formal protoconcept, etc.; and this paper uses the theory of antitone Galois connections in Section 3 to study in detail the distinctive behaviors of the morphisms of the category FCI (formal context interchanges) defined for FCA in Section 3. It is important to emphasize that all the structure of FCA is given by the Birkhoff operators, and therefore, in the authors’ view, it is a categorical imperative that FCI morphisms “interchange” these Birkhoff operators and associated structures; and a consequence of this philosophy is that FCI (and L-FCI and CRSQuantop -FCI) morphisms differ from context morphisms given, for example, in [25]—see the third bullet. It is shown in Section 3, under very general conditions, that the axioms for FCI morphisms are both sufficient and necessary for interchanging formal concepts and formal protoconcepts, structures central to FCA. • It explicitly ties FCA categorically to Chu systems in Section 4 by constructing and studying relationships between the category FCI for FCA and the category Chu2 for Chu systems. And in Section 8, the relationship of FCA to topological systems and other “structured” systems is given a detailed analysis by studying how the objects of FCI are positioned within Chu2 vis-a-vis how topological systems and other structured systems are positioned in Chu2 . It should be mentioned at this point that topological and Chu systems are inherently related to fuzzy sets or lattice-valued mathematics, as seen in [14] and subsequent papers; and so at this juncture implicit links to lattice-valued notions are at hand. • The ties established in the previous two bullets carry over to lattice-valued FCA (L-FCA) for each (L , ⱕ , ⊗) a commutative, residuated semiquantale. L-FCA builds on the L-Birkhoff operators of R. Bˇelohlávek [3,4] and resulting Galois connections and subsequent notions of formal L-concept, formal L-preconcept, formal L-protoconcept. Ties between Galois connection order-theoretics and L-FCA are recorded in Section 5, in which is also developed a detailed analysis comparing lattice-valued FCA as given in this paper with the theory of many-valued contexts given in [25]. It is shown that the family of all many-valued contexts injects into the schema of all formal L-contexts, where L ranges over the class of all commutative, residuated semiquantales. This latter schema motivates the definition of the variable-basis category CRSQuantop -FCI in Section 9 which includes all categories of the form L-FCI introduced in Section 6 and thereby provides a single categorical platform for all of lattice-valued FCA. • The category L-FCI (formal L-context interchanges) for L-FCA is given in Section 6 and with morpshims preserving the all-important L-Birkhoff operators and overlying L-concept structures; and Section 6 gives a detailed analysis of L-FCI morphisms using Galois connection order-theoretics, in which section it is also shown, under very general conditions and parallel to Section 3, that the axioms for L-FCI morphisms are both sufficient and necessary for

54

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

interchanging formal L-concepts and formal L-protoconcepts, structures central to L-FCA. Relationships between L-FCI and both Chu2 and the category Chu L for L-Chu systems are given in detail in Section 7, relationships which are more subtle than their “crisp” counterparts in Section 4. Finally, Section 8 details the relative positioning within Chu L of L-FCI and L-TopSys, the category for L-valued topological systems. Thus the relationships developed in Sections 2–4 are given explicit links to lattice-valued mathematics in Sections 5–8 in addition to those already implicit in the second bullet. • The relationships developed in Section 8 demonstrate the need for both (L-)Chu systems and the categorical study of (L-)FCA: (L-)topological systems cannot categorically accommodate (L-)FCA in a systems context; rather, this accommodation can be done within the category of (L-)Chu systems. This complements the necessity of (L-)Chu systems demonstrated in [13]: (L-)topological systems cannot accommodate all angelic and demonic predicate transformers; rather, this can be done within the category of (L-)Chu systems. Finally, we note that predicate transformers are based upon Galois connections of order-preserving maps [13], while (L-)FCA is based upon Galois connections of order-reversing maps (Section 2); and hence, (L-)Chu systems are justified as a common rubric for Galois connections of both species. • Section 9 highlights and justifies the central emphases of this paper, including the emphasis on the Galois connection of Birkhoff operators associated with a formal context as its only and all-important structure and the need for morphisms in a category for formal contexts to respect this structure. This Section also provides the variablebasis category CRSQuantop -FCI as a supercategory for the schema of L-FCI’s and as a foundation for further variable-basis development of this paper’s ideas. It should be pointed out that a unified theory for FCA related to this paper is being given in [48] using an algebriac variety based approach to topological systems. 2. Formal concept analysis and antitone Galois connections Formal concept analysis (FCA) consists of many well-known methods which may be used for data analysis and knowledge representation. FCA is developed via Galois connections which are defined between powersets and determined by relations between the underlying sets. The underlying sets include a set of objects and a set of attributes which the objects may have. FCA clusters objects in the powerset of objects and clusters attributes in the powerset of attributes, and these clusters are paired by the Galois connection. This pairing is natural with respect to a given relation between the sets of objects and attributes. Definition 2.1. A formal context is an ordered triple (G, M, R), where G is the set of objects, M is the set of attributes, and R is a relation from G to M, i.e., R ⊂ G × M. Anticipating Birkhoff’s fundamental result (Theorem 2.5) that assigns to each formal context a unique Galois connection of Birkhoff operators, it behooves us to first generally define Galois connections of antitone maps and record their important order-theoretic properties. When applied to the Birkhoff operators of a formal context, these properties then give critical insights of a formal context, insights needed throughout this paper. Definition 2.2. A Galois connection is an ordered quadruple ( f, (P, ⱕ ), (Q, ), g) such that (P, ⱕ ) and (Q, ) are partially ordered sets, and f : P → Q and g : Q → P are order-reversing functions such that for each p ∈ P, p ⱕ g f ( p) and for each q ∈ Q, q  f g(q). Definition 2.3. (Alternate Definition) A Galois connection is an ordered quadruple ( f, (P, ⱕ ), (Q, ), g) such that (P, ⱕ ) and (Q, ) are partially ordered sets, and for each p ∈ P and q ∈ Q, p ⱕ g(q) if and only if q  f ( p). Galois connections may be defined with order-reversing or order-preserving functions. They were originally defined for order-reversing functions between powersets by G.D. Birkhoff [8], who called them “polarities”. Subsequently, O. Ore [37] extended Birkhoff’s notion to arbitrary posets and called them “Galois connexions”. It was J. Schmidt [45] who retained the name “Galois connections”, but changed the concept to one with order-preserving functions, in which case the relationship between the order-preserving maps is called “adjunction” in modern parlance [33]. For the work presented in this paper, we use order-reversing functions for Galois connections, as is common in FCA. Sometimes for brevity, we write ( f, g) instead of ( f, (P, ⱕ ), (Q, ), g) for a Galois connection.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

55

The following proposition is well known—see, for example, [17] and [35]—and records basic and important information about the order-theoretic behavior of Galois connections. Proposition 2.4. Let ( f, (P, ⱕ ), (Q, ), g) be a Galois connection. 1. f g f = f and g f g = g. 2. The image points are called fixed points. p ∈ g → (Q) if and only if p = g f ( p). Likewise, q ∈ f → (P) if and only if q = f g(q). 3. P and Q are naturally organized or structured by the fibers of f and g, respectively. Each fiber of f contains exactly one point of g → (Q), and each fiber of g contains exactly one point of f → (P). The image point in each fiber is the largest element of the fiber. 4. The partition of fibers of P has the same partially ordered structure as g → (Q), and the partition of fibers of Q has the same partially ordered structure as f → (P). If E 1 and E 2 are two fibers or equivalence classes, for example, in P, then E 1 ⱕ E 2 if and only if there exist p1 ∈ E 1 and p2 ∈ E 2 such that p1 ⱕ p2 . f → (P) g → (Q) 5. g → (Q) and f → (P) are anti-isomorphic partially ordered sets, and f |g→ (Q) : g → (Q) → f → (P) and g| f → (P) : f → (P)

g → (Q)

f → (P) → g → (Q) are order-reversing bijections. In fact, f |g→ (Q) and g| f → (P) are anti-isomorphic inverses of each other. Hence, the set of fibers in P and the set of fibers in Q are anti-isomorphic partially ordered sets. 6. If P or Q is a [complete] lattice, then so are g → (Q) and f → (P). However, g → (Q) and f → (P) need not be sublattices of P and Q, respectively. Terminology. Because of Proposition 2.4(3), we adopt the following terminology when convenient: a fiber in P or Q is called a leaf, elements in the same leaf are called equivalent, and the largest element of a leaf is called a node. This latter term visually suggests the fact that a leaf attaches to the subset of fixed points in P or Q by its largest element. The following fundamental result from Birkhoff [8,9] is the foundation of much of FCA and L-FCA, in part because it links the critical information of Proposition 2.4 to the ideas of FCA and L-FCA, as shown throughout this paper. Theorem 2.5 (Birkhoff Operators). Let G and M be arbitrary sets and let R ⊂ G × M be a relation. Define H : ℘(G) → ℘(M) and K : ℘(M) → ℘(G) by for S ⊂ G, H (S) = {m ∈ M : g Rm ∀g ∈ S} for T ⊂ M, K (T ) = {g ∈ G : g Rm ∀m ∈ T } (H, ℘(G), ℘(M), K ) is a Galois connection where the orderings on both ℘(G) and ℘(M) are the subset orderings. It should be noted that all Galois connections between powersets arise as Birkhoff operators—see Ore [37]. Definition 2.6. Let (G, M, R) be a formal context. A formal concept of the formal context is an ordered pair (A, B) with A ⊂ G and B ⊂ M such that H (A) = B and K (B) = A. If (A, B) and ( A , B ) are formal concepts of (G, M, R), then ( A, B) ⱕ (A , B ) if A ⊂ A or, equivalently, if B ⊂ B. Definition 2.7. Let K := (G, M, R) be a formal context. The set of all formal concepts of K with the ordering defined in Definition 2.6 is called the concept lattice of K. Theorem 2.8. Let K := (G, M, R) be a formal context and let (H, ℘(G), ℘(M), K ) be the associated Galois connection. The concept lattice of K is a complete lattice, and it is isomorphic to K → (Q) and anti-isomorphic to H → (P). The following definitions of formal preconcept and formal protoconcept come from [54]. Definition 2.9. Let K := (G, M, R) be a formal context. 1. A formal preconcept of the formal context is an ordered pair (C, D) with C ⊂ G and D ⊂ M such that C ⊂ K (D) or, equivalently, D ⊂ H (C). 2. If (C, D) and (C , D ) are formal preconcepts of a formal context, then (C, D)  (C , D ) if C ⊂ C and D ⊂ D . (This partial ordering on formal preconcepts is not an extension of the partial ordering on formal concepts.)

56

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

3. Let K := (G, M, R) be a formal context, and let (C, D) be a formal preconcept. The collection of all formal concepts (A, B) such that (C, D)  (A, B) is a subset of the concept lattice of K and is denoted by Pr econ(C, D). We order the elements of Pr econ(C, D) by the partial ordering on formal concepts, i.e., by ⱕ . We think of formal preconcepts as specifying formal concepts in the sense that formal preconcept (C, D) specifies or “determines” formal concept ( A, B) if (C, D)  (A, B). However, as Pr econ(C, D) is a subset of formal concepts, we use the formal concept partial ordering on Pr econ(C, D). Proposition 2.10. Let K := (G, M, R) be a formal context. 1. If (C, D) be a formal preconcept, then Pr econ(C, D) is itself a complete lattice with (K H (C), H (C)) being the smallest formal concept in Pr econ(C, D) and (K (D), H K (D)) being the largest. 2. For formal preconcept (C, D), every formal concept (A, B) with K H (C) ⊂ A ⊂ K (D) is in Pr econ(C, D). 3. If (C, D) be a formal preconcept, then (C, D) is less than or equal to exactly one formal concept (A, B) in the preconcept partial ordering (i.e., there is a unique formal concept (A, B) with (C, D)  (A, B)) if and only if (A, B) = (K (D), H (C)). Thinking of formal preconcepts as specifying formal concepts, leads to the next proposition. Proposition 2.11. Let K := (G, M, R) be a formal context and let (C, D) and (C , D ) be formal preconcepts with (C, D)  (C , D ). Then Pr econ(C , D ) ⊂ Pr econ(C, D). Thus, the higher a formal preconcept is in the preconcept partial ordering, the more specific or precise it is in specifying formal concepts. However, it is not the case that (C, D)  (C , D ) if and only if Pr econ(C , D ) ⊂ Pr econ(C, D). For example, if D = D and if C and C are not comparable in the subset ordering but are in the same equivalence class, then Pr econ(C, D) = Pr econ(C , D ) but (C, D) and (C , D ) are not comparable in the formal preconcept partial ordering. Since the formal preconcepts may be thought of as specifications for the formal concepts, we propose a pre-order which is defined on the set of formal preconcepts of a formal context and which precisely reflects the thinking that formal preconcepts are specifications for formal concepts. Definition 2.12. Let (C, D) and (C , D ) be formal preconcepts of a formal context K := (G, M, R). Then (C, D)

(C , D ) if Pr econ(C , D ) ⊂ Pr econ(C, D). Proposition 2.13. Let K := (G, M, R) be a formal context and let (C, D) and (C , D ) be formal preconcepts. The following are equivalent: • • • •

(C, D) and (C , D ) are equivalent as formal preconcepts in the pre-order ; (C, D) (C , D ) and (C , D ) (C, D); Pr econ(C, D) = Pr econ(C , D ); C and C are in the same equivalence class of ℘(G) and D and D are in the same equivalence class of ℘(M).

Definition 2.14. Let (G, M, R) be a formal context. A formal protoconcept of the formal context is a formal preconcept (C, D) such that Pr econ(C, D) contains exactly one formal concept. Theorem 2.15. Let (G, M, R) be a formal context, let (H, K ) be the associated Galois connection of Birkhoff operators and let C ⊂ G, D ⊂ M. The following are equivalent: 1. 2. 3. 4. 5. 6.

(C, D) is a formal protoconcept. Pr econ(C, D) contains exactly one formal concept. C, D are members of corresponding leaves under anti-isomorphisms between K → (℘(M)) and H → (℘(G)). (K (D), H (C)) is a formal concept of (G, M, R). K H (C) = K (D). H K (D) = H (C).

Statement (4) of Theorem 2.15 justifies the term “protoconcept”; and in practice, (5) and (6) seem the most convenient to apply.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

57

3. Category FCI for formal contexts To facilitate additional mathematical investigations in FCA, we want to define a category whose objects are formal contexts. Questions which immediately come to mind include what are the morphisms of such a category, does the category have a base category with a natural forgetful functor, and what properties does the category have? In the previous paragraph, we phrased the questions as if there were only one possible category. Of course, there may be several natural and useful categories which have formal contexts as their objects. As we address the first and most immediate question, which is what are the morphisms of this category, we ask ourselves what properties or characteristics do formal contexts have and what properties should the morphisms preserve. Interestingly, though a formal context is defined in terms of two sets and a relation between them, the structure of the formal context and its important aspects are the order-theoretic properties of the associated Galois connection at the powerset level given by Theorem 2.5 in concert with the other results of Section 2. Thus, given two formal contexts K1 = (G 1 , M1 , R1 ) and K2 = (G 2 , M2 , R2 ), a morphism from K1 to K2 needs to respect the Galois connections (H1 , ℘(G 1 ), ℘(M1 ), K 1 ) and (H2 , ℘(G 2 ), ℘(M2 ), K 2 ), determined by K1 and K2 , respectively. One way of defining morphisms from K1 to K2 is to define the morphisms as pairs of functions ( f, g) such that ( f, g) : K1 → K2 if f : ℘(G 1 ) → ℘(G 2 ) and g : ℘(M1 ) → ℘(M2 ) with H2 ◦ f = g ◦ H1 and f ◦ K 1 = K 2 ◦ g. The above category, which we dub GalConn for future reference, closely parallels the category GAL previously defined in [36]: the primary difference between GalConn and GAL is in the objects—the Galois connections associated with GalConn objects comprise order-reversing operators, but in GAL the analogous operators are order-preserving. The reader should be aware that in [48] occurs the category GalCon, which, despite its name being similar to GalConn, has very different morphisms. Now let ( f, g) : K1 → K2 be a GalConn morphism. In [36] is considered the case where f and g may be arbitrary functions and the case where they are order-preserving functions. We assume, unless stated otherwise, that f and g are arbitrary functions. The first three of the following four statements are consequences of Proposition 2.04 of [36], using the general fact that (G, M, R) ∈ |GalConn| ⇔ (℘(G), H, K , ℘(M)op ) ∈ |GAL|, while the fourth statement follows from the discussion of order-preserving morphism components in [36]: • If A1 and A 1 are in the same equivalence class of ℘(G 1 ), then f (A1 ) and f (A 1 ) must be in the same equivalence class of ℘(G 2 ). Similarly, if B1 and B1 are in the same equivalence class of ℘(M1 ), then g(B1 ) and g(B1 ) must be in the same equivalence class in ℘(M2 ). • f and g take fixed subsets to fixed subsets. • Formal protoconcepts in K1 are mapped to formal protoconcepts in K2 ; i.e., if (C, D) is a formal protoconcept in K1 , then ( f (C), g(D)) is a formal protoconcept in K2 . • If f, g are order-preserving on nodes, then formal preconcepts in K1 are mapped to formal preconcepts in K2 ; i.e., if (C, D) is a formal preconcept in K1 , then ( f (C), g(D)) is a formal preconcept in K2 . It is noted that a powerset has a natural ordering—the subset ordering, and this ordering is important in FCA. Thus, in some applications of FCA, there appears the additional property that f and g are order-preserving. In this paper, we are, however, interested in a different category of formal contexts. For reasons which will become clear later and are related to Chu and topological systems, we want the second function in an ordered pair of a morphism ( f, g) : K1 → K2 to be defined from ℘(M2 ) to ℘(M1 ), i.e., we want g : ℘(M2 ) → ℘(M1 ). Desirable properties, answering to those itemized above for GalConn, should include the following: • f and g respect Galois connections’ generated equivalence classes, i.e., elements in one equivalence class must be mapped into the same equivalence class, so that each of f and g respects leaves—cf. Theorem 3.3(1) below;

58

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

• f and g map fixed subsets to fixed subsets and, further, “interchange” formal concepts of the underlying formal contexts—cf. Theorem 3.3(2); • f and g “interchange” formal protoconcepts of the underlying formal contexts—cf. Theorem 3.3(4); and • if f and g are order-preserving on nodes, then they “interchange” formal preconcepts of the underlying formal contexts—cf. Theorem 3.3(6). Definition 3.1. The category FCI of formal context interchanges comprises objects which are formal contexts K = (G, M, R) and morphisms ( f, ) : K1 → K2 such that f : ℘(G 1 ) → ℘(G 2 ) and op : ℘(M2 ) → ℘(M1 ) are maps satisfying H1 = op ◦ H2 ◦ f and K 2 = f ◦ K 1 ◦ op , where (H1 , ℘(G 1 ), ℘(M1 ), K 1 ) and (H2 , ℘(G 2 ), ℘(M2 ), K 2 ) are the Galois connections respectively determined by K1 and K2 . Note that we have replaced the g : ℘(M2 ) → ℘(M1 ) in the above discussion by op : ℘(M2 ) → ℘(M1 ), where  is a morphism in Setop , so that a ground category for FCI is Set × Setop and the forgetful functor applied to K, when K = (G, M, R), yields (℘(G), ℘(M)). This is done to facilitate relationships between formal contexts and topological and Chu systems in subsequent sections. It should be emphasized that Definition 3.1 does not require that either f or op be order-preserving. Further, the commutivities of the display in Definition 3.1 are not sufficient to imply that f or op preserves order; it is straightforward to construct examples justifying this claim. However, it is interesting to observe these commutivities control to a certain degree the behavior of f and op with respect to order, so f and op cannot change the ordering “randomly” and independently of each other: each violation of the ordering by f must be matched by a complementary behavior by op which “masks” in the first commutivity the order-violation by f, and vice versa with respect to the second commutivity. These ideas are clarified in Theorem 3.3(5). The four properties listed immediately above Definition 3.1, and more besides, are demonstrated in Theorem 3.3; and Theorem 3.4, giving converses for both 3.3(1–4) and the results itemized above for GalConn morphisms, demonstrates that the definition of FCI [GalConn] morphisms is a formal necessity for any categorial approach which uses “forwardand-backward” [“forward-and-forward”] powerset level maps and which “interchanges” [preserves] the formal analytic structure of formal contexts—see Section 9 below. But we first need the following definition: Definition 3.2. A mapping h from a preordered set (X, ⱕ ) to a preordered set (Y, ⱕ ) detects order if ∀x1 , x2 ∈ X , h(x1 ) ⱕ h(x2 ) ⇒ x1 ⱕ x2 . Theorem 3.3. Let ( f, ) : K1 → K2 be a morphism in FCI. Then the following hold: 1. f and op map equivalent subsets of objects and attributes, respectively, to equivalent subsets of objects and attributes, so that each of f and op respects leaves, respectively, from ℘(G 1 ) to ℘(G 2 ), from ℘(M2 ) to ℘(M1 ). 2. (a) f and op map fixed subsets of ℘(G 1 ) and ℘(M2 ), respectively, to fixed subsets of ℘(G 2 ) and ℘(M1 ), so that each of f and op respects leaf nodes. Further, (b) ( f, ) “interchanges” formal concepts in the sense that ∀C ∈ ℘(G 1 ), D ∈ ℘(M2 ) with C, D leaf nodes, (C, op (D)) is a formal concept if and only if ( f (C), D) is a formal concept. 3. Each of f and op is a bijection on leaf nodes. 4. If C1 is a subset of objects in K1 and D2 is a subset of attributes in K2 , then (C1 , op (D2 )) is a formal protoconcept of K1 if and only if ( f (C1 ), D2 ) is a formal protoconcept of K2 ; so that ( f, ) “interchanges” formal protoconcepts. 5. One of the maps f, op detects ordering of leaf nodes if and only if the other map preserves ordering of leaf nodes. 6. f and op preserve orderings of leaf nodes if and only if ( f, ) “interchanges” formal preconcepts in this sense: for each C1 a subset of objects in K1 and D2 a subset of attributes in K2 , (C1 , op (D2 )) is a formal preconcept of K1 if and only if ( f (C1 ), D2 ) is a formal preconcept of K2 . K → (℘(M2 )) 7. f and op preserve orderings of leaf nodes if and only if both f |(K21 op )→ (℘(M2 )) : (K 1 op )→ (℘(M2 )) → K 2→ (℘(M2 )) H → (℘(G 1 ))

and op |(H12 f )→ (℘(G 1 )) : (H2 f )→ (℘(G 1 )) → H1→ (℘(G 1 )) are order isomorphisms.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

59

Proof. Each assertion ultimately follows from the commutivity conditions H1 = op ◦ H2 ◦ f and K 2 = f ◦ K 1 ◦ op of Definition 3.1. Ad (1). Let A1 , A 1 ∈ ℘(G 1 ) such that H1 (A1 ) = H1 (A 1 ). Then op (H2 ( f (A1 ))) = op (H2 ( f (A 1 ))), f (K 1 (op (H2 ( f (A1 ))))) = f (K 1 (op (H2 ( f (A 1 ))))), K 2 H2 ( f (A1 )) = K 2 H2 ( f (A 1 )), H2 K 2 H2 ( f (A1 )) = H2 K 2 H2 ( f (A 1 )), H2 ( f (A1 )) = H2 ( f (A 1 )), showing that f preserves equivalence classes. And the proof is similar that op does this as well. Ad (2). Let B ∈ ℘(M2 ) such that B = H2 K 2 (B). Then K 2 (B) = f (K 1 (op (B))), op (H2 K 2 (B)) = op (H2 ( f (K 1 (op (B))))), op (B) = op (H2 ( f (K 1 (op (B))))), op (B) = H1 K 1 (op (B)), showing that op preserves fixed subsets of M2 ; and the proof is similar that f preserves fixed subsets of G 1 . Now to see that ( f, ) interchanges formal concepts, let A ∈ ℘(G 1 ), D ∈ ℘(M2 ) with A, D leaf nodes. For necessity, assume that (A, op (D)) is a formal concept. Then H1 (A) = op (D) and A = K 1 (op (D)). Applying f and then the first commutivity, we have f (A) = f (K 1 (op (D))) = K 2 (D), and note that f (A) = K 2 (D) is half of what is needed to say that ( f (A), D) is a formal concept. Now applying H2 to this half and noting D is a leaf node yield that H2 ( f (A)) = H2 K 2 (D) = D, so that H2 ( f (A)) = D, the other half of what is needed to say that ( f (A), D) is a formal concept. The proof of sufficiency is dual and uses the second commutivity. Ad (3). We note that Proposition 2.4(5), in concert with the preceding statements of Proposition 2.4, implies that each of H1 , K 1 , H2 , K 2 is bijective on the leaf nodes of its domain. Now it is a well-known fact that the first arrow in the factorization of a monomorphism must be a monomorphism; and so it follows from the first commutivity H1 = op ◦ H2 ◦ f and the injectivity of H1 on leaf nodes that f is injective on leaf nodes. It also a well-known fact that the last arrow in the factorization of an epimorphism must be an epimorphism; and so it follows from the second commutivity K 2 = f ◦ K 1 ◦ op and the surjectivity of K 2 on leaf nodes that f is surjective on leaf nodes. And so f is a bijection on leaf nodes. The proof that op is a bijection on leaf nodes is similar. Ad (4). Let C1 ⊂ G 1 and let D2 ⊂ M2 such that (C1 , op (D2 )) is a formal protoconcept. Then H1 K 1 (op (D2 )) = H1 (C1 ) by Theorem 2.15(6). Beginning with H1 K 1 (op (D2 )) = H1 (C1 ),

60

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

we have H1 K 1 (op (D2 )) = op (H2 ( f (C1 ))), f (K 1 H1 K 1 (op (D2 ))) = f (K 1 (op (H2 ( f (C1 ))))), f (K 1 (op (D2 ))) = f (K 1 (op (H2 ( f (C1 ))))), K 2 (D2 ) = K 2 H2 ( f (C1 )). Hence, ( f (C1 ), D2 ) is a formal protoconcept by Theorem 2.15(5). Ad (5). Statement (5) actually encompasses two biconditionals, one of which is proved; the other biconditional may be proved similarly. Assume that f detects order on leaf nodes, and let D1 ⊂ D2 as nodes in ℘(M2 ). Since K 2 is anti-isomorphic on leaf nodes, then f K 1 op (D2 ) = K 2 (D2 ) ⊂ K 2 (D1 ) = f K 1 op (D1 ). Since f detects order on leaf nodes, then K 1 op (D2 ) ⊂ K 1 op (D1 ). Hence, by K 1 being anti-isomorphic on leaf nodes, op (D1 ) ⊂ op (D2 ), and op preserves the ordering on leaf nodes. For the converse direction, assume that op is isotone on the leaf nodes, and let A1 , A2 be nodes in ℘(G 1 ) with f (A1 ) ⊂ f (A2 ). The antitonicity of H2 and the isotonicity of op yield H1 (A1 ) = op (H2 ( f (A1 ))) ⊃ op (H2 ( f (A2 ))) = H1 (A2 ). Finally, H1 being anti-isomorphic on leaf nodes implies A1 ⊂ A2 . Ad (6). To show necessity in Statement (6), it is assumed that both f, op preserve order on leaf nodes. It must then be shown that (C1 , op (D2 )) is a formal preconcept of K1 if and only if ( f (C1 ), D2 ) is a formal preconcept of K2 . For the necessity of the consequent of Statement (6), let (C1 , op (D2 )) be a formal preconcept of K1 . Then C1 ⊂ K 1 (op (D2 )), and therefore K 1 H1 (C1 ) ⊂ K 1 H1 K 1 (op (D2 )) = K 1 (op (D2 )). Now the isotonicity of f on leaf nodes and the first commutivity condition of the FCI morphism ( f, ) together yield f (K 1 H1 (C1 )) ⊂ f (K 1 (op (D2 ))) = K 2 (D2 ). Finally, we appeal to statements (1) and (2) above that f preserves both leaves and leaf nodes to conclude that f (C1 ) ⊂ f (K 1 H1 (C1 )), which yields that f (C1 ) ⊂ K 2 (D2 ). It follows that ( f (C1 ), D2 ) is a formal preconcept of K2 . To show the sufficiency of the consequent of Statement (6), let ( f (C1 ), D2 ) be a formal preconcept of K2 , which means that D2 ⊂ H2 f (C1 ). It follows that H2 K 2 (D2 ) ⊂ H2 K 2 H2 f (C1 ) = H2 f (C1 ), and therefore, op H2 K 2 (D2 ) ⊂ op H2 f (C1 ) = H1 (C1 ). Since K 2 (D2 ) = K 2 H2 K 2 (D2 ), then op (D2 ) ⊂ op H2 K 2 (D2 ), and hence, op (D2 ) ⊂ H1 (C1 ). Thus, (C1 , op (D2 )) is a formal preconcept of K1 , ( f, ) interchanges formal preconcepts, and the proof of necessity in Statement (6) is complete. For sufficiency in Statement (6), we assume that ( f, ) interchanges formal preconcepts. It must be shown that each of f, op preserves the ordering of leaf nodes. We apply Proposition 2.4 and Statement (3) above to note that each

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

61

of H1 , K 1 , H2 , K 2 , f, op is bijective on leaf nodes. Now to show that op preserves the ordering of leaf nodes, let D ⊂ D in ℘(M2 ). Then the bijectivities of H2 , f imply that ∃!C ∈ ℘(G 1 ) with D = H2 ( f (C)). Immediately, we have D ⊂ H2 ( f (C)). Now the antitonicity of K 2 applied to the previous line, along with the fact that K 2 H2 ( f (C)) is the node of the leaf in ℘(G 2 ) containing f (C), together yield that f (C) ⊂ K 2 H2 ( f (C)) ⊂ K 2 (D ). The assumption that ( f, ) interchanges formal preconcepts now implies that C ⊂ K 1 (op (D )), which, using the antitonicity of H1 and the fact that H1 K 1 (op (D )) is the node of the leaf containing op (D ), implies that op (D ) ⊂ H1 K 1 (op (D )) ⊂ H1 (C). But the equation D = H2 ( f (C)) and the first commutivity condition together yield that op (D) = op (H2 ( f (C))) = H1 (C). The two previous displays together say that op (D ) ⊂ op (D), completing the proof that op is order-preserving on leaf nodes. The proof that f is order-preserving on leaf nodes is similar. Thus sufficiency in Statement (6) holds. Ad (7). Follows from (3) and (5).  It should be pointed out that a number of different proofs can be given for Theorem 3.3 since some statements of Theorem 3.3 are consequences of various combinations of other statements. Consequently, different proofs can also be given for the following converse of Theorem 3.3(1–4). Theorem 3.4 (Converse to Theorem 3.3(1–4)). Let K1 = (G 1 , M1 , R1 ), K2 = (G 2 , M2 , R2 ) be formal contexts. 1. If ( f, ) : (℘(G 1 ), ℘(M1 )) → (℘(G 2 ), ℘(M2 )) is a Set × Setop morphism satisfying the conditions of Theorem 3.3(1–4), then ( f, ) is a morphism in FCI. 2. If ( f, g) : (℘(G 1 ), ℘(M1 )) → (℘(G 2 ), ℘(M2 )) is a Set ×Setop morphism satisfying the consequences of Proposition 2.04 [36] listed above in the sixth paragraph of this section, then ( f, ) is a morphism in GalConn. Proof. The proof of (2) is parallel to that of (1) and left to the reader. Various proofs of (1) can be given; and our proof of (1) only uses Theorem 3.3(2(a),4). Thus, the goal is to show that Theorem 3.3(2(a),4) implies the two commutivity conditions of Definition 3.1. We prove only the first commutivity condition; the proof of the second commutivity is symmetric and left to the reader. Let C ∈ ℘(G 1 ). It needs to be shown that H1 (C) = op (H2 ( f (C))). Now put D ∈ ℘(M2 ) by D = H2 ( f (C)). It follows that D is a node and that K 2 (D) = K 2 H2 ( f (C)).

62

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

But this says that ( f (C), D) is a protoconcept in K2 . So Theorem 3.3(4) now implies that (C, (D)) is a protoconcept in K1 ; and hence H1 (C) = H1 K 1 (op (D)). It follows that H1 (C) is a node of the leaf in ℘(M1 ) containing op (D). But we know that D is a node, so then op (D) is a node by Theorem 3.3(2)(a). This forces H1 (C) = op (D) = op (H2 ( f (C))), completing the proof.  Theorem 3.3 shows that FCI is a categorical framework for FCA whose morphisms honor the associated Galois connections and their generated equivalence relations, fixed subsets, formal concepts, and formal protoconcepts; and Theorem 3.4 shows that honoring such structures forces the commutivities defining FCI morphisms; cf. Section 9. It should be emphasized that the bijectivity of f and op for FCI morphism ( f, ) generally occurs only on the leaf nodes (examples illustrating this are easy to construct and left to the reader). This indicates the richness of FCI morphisms as well as their distinctive character. This does not appear to be the case for the morphisms of GalConn, a fact motivating the ongoing study of GalConn morphisms by the authors. The richness of FCI morphisms is enhanced by the distinctive order properties of these component maps f and op , as seen in Theorem 3.3(5), forced by the commutivity conditions of Definition 3.1. Further, the special role played by those FCI morphisms with component maps f and op that preserve order on the leaf nodes is that these are precisely the morphisms which honor formal preconcepts. Another aspect of the richness of FCI morphisms is indicated by this observation: even if one of the component maps is order-preserving on the leaf nodes, it is generally not an orderisomorphism on the leaf nodes (which means it is not order-detecting and its inverse map is not isotone)—again, the examples are easy to construct and left to the reader. Finally, we point out that component maps f and op preserving order on the leaf nodes does not imply that these maps preserve order on all their respective domains. The following example justifies this observation. Example 3.5. Let K1 := (G 1 , M1 , R1 ) be given by G 1 = {x, y, z},

M1 = {a},

R1 = ⭋,

and put K2 = K1 . To define ( f, ) : K1 → K2 , we first define op := id : ℘(M2 ) → ℘(M1 ). Now put f : ℘(G 1 ) → ℘(G 2 ) according to the following table: V

f (V )

⭋ ⭋ {x} {x, y} {y} {y} {z} {z} {x, y} {x} {x, z} {x, z} {y, z} {y, z} {x, y, z} {x, y, z} It can be checked that ( f, ) is an FCI morphism from K1 to K2 for which the component maps f and op each preserves leaf node order—the nodes of ℘(G 1 ), ℘(G 2 ) are ⭋ and {x, y, z}, and the nodes of ℘(M2 ), ℘(M1 ) are ⭋ and {a}. But f does not preserve order overall: {x} ⊂ {x, y}, but f {x} = {x, y} ⊂ / {x} = f {x, y}.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

63

4. Formal contexts (FCI) and Chu systems (Chu2 ) A Chu space or Chu system is a well-known concept [34,38] which can be built from relationships between objects in one set and their attributes in another set. Interestingly, Chu systems underly both topological systems [53]—systems in which the second set is a locale of predicates and the relation then matches objects of the first set with their predicates in the locale—as well as systems arising from predicate transformers in programming semantics [11]. This section reviews Chu systems and begins the development of their relationships to formal contexts, a development which is extended and continued in Sections 6 and 7 below. We point out that even in the seemingly crisp setting of the current section, the relationships infra between contexts and systems—given that topological systems are inherently fuzzy [14]—already link traditional contexts to fuzzy sets, a linkage explicitly multiplied in Sections 6 and 7. The next definition, discussion, and subsequent proposition are from [34,38] and [11]. Definition 4.1. A Chu space or Chu system is a triple (X, A, ), where (X, A) ∈ |Set × Setop | and  is a satisfaction relation from X to A, i.e.,  ⊂ X × A is a relation from X to A. The set A is the set of predicates. Chu transforms between Chu systems are ordered pairs ( f, ) : (X, A, 1 ) → (Y, B, 2 ) with ( f, ) ∈ Set × Setop , f : X → Y a set function and  : A → B a Setop morphism satisfying the adjointness property that for all x ∈ X and all b ∈ B, f (x) 2 b if and only if x 1 op (b). The category Chu2 comprises all Chu systems and Chu transforms, along with the compositions and identities inherited from Set × Setop . We refer to Set × Setop as the ground category for Chu2 . In this paper we strongly prefer the terminology “Chu system” over “Chu space” for these reasons: the system terminology emphasizes the connection with topological systems, measurable systems, systems arising from predicate transformers, and other types of systems—see [11] and Section 8; the term “space” suggests a kind of topological setting, but it is easily seen that Chu2 is not a topological category over its ground category; and the system terminology allows us to carefully distinguish between Chu2 and the “Chu extent spaces” which it induces, spaces which are indeed part of a topological construct. As just mentioned, closely associated with Chu systems and Chu transforms are “Chu extent spaces” and “Chu continuous” mappings. Given a Chu system (X, A, ), there is a mapping ext : A → ℘(X ) defined by ext(a) = {x ∈ X : x  a}, along with the Chu extent space (X, ext → (A)). Proposition 4.2. If ( f, ) : (X, A, 1 ) → (Y, B, 2 ) is a Chu2 morphism, then f : (X, ext → (A)) → (Y, ext → (B)) has the property that ∀V ∈ ext → (B),

f ← (V ) ∈ ext → (A).

The proposition justifies saying that the map f is Chu-continuous. The category ChuTop comprises Chu topological spaces (X, A) with A ⊂ ℘(X ), along with Chu continuous mappings f between the ground sets of spaces (X, A), (Y, B) such that ( f ← )→ (B) ⊂ A. It can be shown that ChuTop is topological over Set with respect to the expected forgetful functor: the proof parallels that of the topologicity of Top over Set with respect to the usual forgetful functor. As pointed out by a referee, ChuTop can be redescribed as a “functor costructured category” (5.40, 5.43 of [1]). Letting Q : Setop → Set be the covariant “preimage” powerset functor given by Q(X ) = ℘(X ), Q(( f : X ← Y )op : X → Y ) = f ← : ℘(X ) → ℘(Y ),

64

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

then the functor costructured category (Spa(Q))op is defined as follows (5.44 of [1]): objects are pairs (X, A) with A ⊂ ℘(X ), and morphisms f : (X, A) → (Y, B) with f : X → Y a function such that (Q( f op : X ← Y ))→ (B) ⊂ A. Clearly, (Spa(Q))op = ChuTop, linking costructured categories to systems and FCA. It should be noted that the above proposition and discussion construct the extent functor E xt : Chu2 → ChuTop. Links of these notions to infratopological systems and their associated extent spaces – which are infratopological spaces – are introduced in the next example. Example 4.3. 1. Let ( f, (P, ⱕ ), (Q, ), g) be a Galois connection with order-reversing maps. Then ( f, g op ) : (P, P, ⱕ ) → (Q, Q, op ) is a Chu2 morphism. (a) For the Chu system, (P, P, ⱕ ) with p ∈ P, ext( p) = {x ∈ P : x ⱕ p}, i.e., ext( p) is the downset or principal ideal ↓( p) determined by p in P. (b) To modify (a), assume that P is a complete lattice. Then (P, P, ⱕ ) is an infra-topological system in the sense of [13] and Section 8 below—its associated extent space is an infratopological space. 2. Let ( f, (P, ⱕ ), (Q, ), g) be a Galois connection with order-preserving maps. Then ( f, g op ) : (P, P, ⱕ ) → (Q, Q, ) is a Chu2 morphism. Definition 4.4. Let K = (G, M, R) be a formal context. 1. 2. 3. 4. 5. 6.

Define  pr e ⊂ ℘(G) × ℘(M) by C  pr e D if (C, D) is a formal preconcept in K. Define  pr o ⊂ ℘(G) × ℘(M) by C  pr o D if (C, D) is a formal protoconcept in K. Define con ⊂ ℘(G) × ℘(M) by C con D if (C, D) is a formal concept in K. Define IK pr e = (℘(G), ℘(M),  pr e ). Define IK pr o = (℘(G), ℘(M),  pr o ). Define IKcon = (℘(G), ℘(M), con ).

Example 4.5. Let ( f, ) : K1 → K2 be a morphism in FCI. pr e

pr e

1. According to Theorem 3.3(6), ( f, ) : K1 → K2 is a Chu2 morphism if and only if f and op are order-preserving maps. For a Chu system (℘(G), ℘(M),  pr e ) and for D ⊂ M, ext(D) = {C ⊂ G : C ⊂ K (D)} because for each such C, (C, D) is a preconcept of K = (G, M, R). The ext(D) is interesting because there may be C1 , C2 ∈ ext(D) such that C1 and C2 are in the same leaf but are not comparable as subsets of G and such that (C1 , D) and (C2 , D) are not comparable in the partial order  on formal preconcepts. However, when C1 and C2 are in the same leaf, then (C1 , D) and (C2 , D) are comparable in pre-order (Definition 2.12) on formal preconcepts. pr o pr o 2. According to Theorem 3.3(4), ( f, ) : K1 → K2 is a Chu2 morphism. pr o For a Chu system (℘(G), ℘(M),  ) with D ⊂ M, ext(D) = {C ⊂ G : K H (C) = K (D)}, i.e., all the subsets of G which are in the leaf of K (D). 3. Similarly, ( f, ) : K1con → K2con is a Chu2 morphism. For a Chu system (℘(G), ℘(M), con ) with D ⊂ M,  {K (D)}, if D is a leaf node ext(D) = ⭋, otherwise The next theorem insures a bountiful supply of meaningful objects and morphisms in FCI, and for this theorem it is important to have the subcategory Chu2em of Chu2 comprising all objects of Chu2 , together with all morphisms ( f, ) which are epi-monos, i.e., f is epi (surjective) in Set and  is mono in Setop (so that op is surjective in Set). Theorem 4.6. Iem : Chu2em → FCI is an embedding, where Iem (X, A, ) = (X, A, ). Iem [( f, ) : (X, A, 1 ) → (Y, B, 2 )] = ( f → , ((op )→ )op ) : (X, A, 1 ) → (Y, B, 2 ).

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

65

It is also the case that Jem : Chu2em → FCI is a covariant functor, where Jem (X, A, ) = (X, A, ).   Jem ( f, ) : (X, A, 1 ) → (Y, B, 2 ) = ( f ⇒ , ((op )⇒ )op ) : (X, A, 1 ) → (Y, B, 2 ). and f ⇒ := K 2 ◦ H2 ◦ f → and (op )⇒ := H1 ◦ K 1 ◦ (op )→ , letting (H1 , K 1 ) be the Galois connection induced by (X, A, 1 ) and (H2 , K 2 ) be the Galois connection induced by (Y, B, 2 ). Proof. The proof that Iem is an embedding follows from the proof of Theorem 7.3 by setting L = 2.  As pointed out by an example from a referee, Jem cannot, in general, be an embedding. Let X = {x1 , x2 }, and let f : X → X by f (x1 ) = x2 and f (x2 ) = x1 . Now (id X , ⭋op ) : (X, ⭋, ⭋) → (X, ⭋, ⭋) and ( f, ⭋op ) : (X, ⭋, ⭋) → (X, ⭋, ⭋) are distinct morphisms in Chu2em . However, since ℘(⭋) = {⭋}, then K H : ℘(X ) → ℘(X ) is a constant map. It follows that (id X )⇒ = K H (id X )→ = K H f → = f ⇒ . Hence, Jem (id X , ⭋op ) = Jem ( f, ⭋op ) and Jem is not an embedding. The underlying reason why Jem is not an embedding is that it cannot distinguish between functions which only differ within leaves. The necessity for the subcategory Chu2em can be seen in the proof that Iem ( f, ) is an FCI morphism: the functions f and op need to be surjective because the adjointness condition for ( f, ), along with the relations 1 and 2 , may involve elements in Y and A, respectively, which are not in the images of arbitrary functions f and op , respectively. It should be pointed out that Iem is not an isomorphism. Since the image operators of surjective maps are surjective powerset operators and FCI has morphisms (g, ) for which g or op need not be surjective, it follows that Iem cannot generate such maps. Even if we were to restrict the codomain ofIem to FCIem , it would still be the case that Iem is not an isomorphism: Iem produces (g, ), where g, op preserve and hence ⊂; but FCIem has morphisms (g, ) for which neither g nor op preserves ⊂. Let K = (G, M, R) be a formal context. Define IK = (℘(G), ℘(M),  pr o ), where  pr o is the relation defined above such that C  pr o D if and only if (C, D) is a formal protoconcept. Additionally, let K1 = (G 1 , M1 , R1 ) and K2 = (G 2 , M2 , R2 ) be formal contexts with ( f, ) : K1 → K2 a formal context morphism. We claim that ( f, ) : IK1 → IK2 is a Chu transform; and to show this, let C1 ∈ ℘(G 1 ) and D2 ∈ ℘(M2 ). It follows from Theorem 3.3(4) that ( f (C1 ), D2 ) is a formal protoconcept in K2 if and only if (C1 , op (D2 )) is a formal protoconcept in K1 . Hence, pr o pr o f (C1 ) 2 D2 if and only if C1 1 op (D2 ), and therefore ( f, ) : IK1 → IK2 is a Chu transform. Thus, we have a functor I : FCI → Chu2 such that I(K) = IK and I(( f, ) : K1 → K2 ) = ( f, ) : IK1 → IK2 . Similarly, we can work with the category FCIop whose objects are FCI-objects and whose morphisms are FCImorphisms which are order-preserving on leaf nodes. We can define IK pr e = (℘(G), ℘(M),  pr e ), where  pr e is the relation defined above such that C  pr e D if and only if (C, D) is a formal preconcept. Let K1 = (G 1 , M1 , R1 ) and K2 = (G 2 , M2 , R2 ) be formal contexts with ( f, ) : K1 → K2 a formal context morphism. From Theorem 3.3(6), pr e pr e ( f, ) : IK1 → IK2 is a Chu transform. Thus, we have a functor Iop : FCIop → Chu2 such that Iop (K) = IK pr e and pr e

Iop (( f, ) : K1 → K2 ) = ( f, ) : IK1 In fact, we have the following.

pr e

→ IK2 .

66

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Theorem 4.7. I : FCI → Chu2 and Iop : FCIop → Chu2 are embeddings. The verification of I being an embedding in Theorem 4.7 follows from Corollary 7.5 and subsequent discussion. It should be pointed out that I is not an isomorphism: it is obvious from Remark 8.19 that I is not object onto. In [11], lattice-valued extensions to Chu systems are studied as a rubric for systems arising from lattice-valued predicate transformers in program semantics. The current work goes on to investigate analogous lattice-valued extensions of both formal contexts and concepts and the consequences of lattice-valued extensions of FCI for lattice-valued Chu systems and for FCA. Thus, the current work will also include possible applications and relationships of lattice-valued FCA to programming semantics. 5. Formal L-contexts Lattice-theoretic preliminaries for this and subsequent sections of this paper are now discussed. Throughout the remainder of this paper, and unless stated otherwise, (L , ⱕ , ⊗) or L is a commutative, residuated semiquantale with tensor product ⊗. This means, precisely, the following standing assumptions: 1. (L , ⱕ , ⊗) is a semiquantale [42], i.e., (L , ⱕ ) is a complete lattice and the tensor ⊗ : L × L → L is a binary operation. The notion of semiquantale makes no additional assumptions on ⊗. 2. It is assumed in the remainder of this paper that the tensor ⊗ distributes from both sides across arbitrary joins, and so there is a residuum or implication → associated with ⊗ as a consequence of the Adjoint Functor Theorem for posets [33]. Since ⊗ preserves arbitrary joins in both arguments, it is order-preserving in both arguments and preserves bottom in both arguments. 3. It is assumed in the remainder of this paper that ⊗ is commutative as a binary operation. In the sequel, L is sometimes also assumed consistent, namely that ⊥  , or equivalently, that |L| ⱖ 2; and it is sometimes also assumed that  is an identity for ⊗ (the condition of being strictly two-sided). Standing assumptions (1,2) give the definition of a residuated semiquantale, a notion lacking the requirement of associativity of ⊗ and appropriate for this paper since no associativity of ⊗ is needed in this and subsequent sections. It is noted that residuated semiquantales are more general than quantales as defined in standard references such as [44] and more general than residuated lattices as defined in standard references such as [22]. It should also be pointed out that the only categorical difference between a residuated semiquantale and a complete residuated lattice with respect to objects is that, in the former, associativity is not required; and the categorical difference with respect to morphisms is that, in the former, morphisms are mappings which preserve arbitrary joins and the tensor, while in the latter, morphisms are mappings preserving the residuum. There has just appeared in the literature the notion of a residuated groupoid [23,21], which, except for completeness, is the same as a residuated semiquantale with respect to objects, but whose associated morphisms are mappings preserving the residuum. Finally, standing assumption (3) above deserves a few comments. It must be noted that the notion of a residuated lattice, in the fundamental and recent references developing and standardizing such lattices [19–23,30,32,50], does not assume that the tensor is commutative. It would be beneficial for all workers in lattice-valued mathematics to follow this standardized vocabulary, as has already been done by many such workers—see [16,18–23,28–30,32,50] and their respective bibliographies. In this context, it is important to note the examples adduced in these references which document the necessity of not assuming commutivity of the tensor operation; e.g., see the introduction of [16]. Returning now to the issue of formal L-contexts, this section motivates the need for formal L-contexts with a series of examples from two sources: surveys and data mining. This is followed by the definition of formal L-contexts, which is compared in detail with the many-valued contexts of [25] and shown in a certain sense to include the latter. Finally, the L-Birkhoff operators for formal L-contexts are given, and then an inventory is made of the resulting lattice-valued extensions of the notions presented in Section 2. In [52], a number of illustrations are given of formal contexts represented by incidence matrices and their resulting formal concept lattices. One of these illustrations—see Figs. 16 and 17 on p. 22—concerns a survey which asks survey-takers to assign various musical attributes (“well-structured”, “dramatic”, etc.) to various classical compositions. The first composition in the incidence matrix is Beethoven’s Romance in F-major for violin and orchestra, a composition which one of the authors of this paper knows intimately. Now Wille’s incidence matrix assigns a blank or 0 for the attributes “dramatic”, “lively”, “fast”; but, despite the lyricism of the work, there are indeed parts which are dramatic

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

67

(the violin jumping from A to g2 and from G1 to f2 toward the beginning of the development), others which are lively (the violin syncopating 3 against the orchestra’s 4 in the development), and yet others which are fast (the violin’s fast runs at the end of the development). Thus the incidence matrix surveying one of this paper’s authors could more accurately reflect his opinion if truth values other than 0 (or ⊥) and 1 (or ) could be marked for these three attributes for this particular composition. But this means that the incidence relation R is a mapping from G × M to some L, giving rise to the notion of L-formal contexts. It should also be pointed out that if more than one person takes the musical survey indicated in Figures 16 and 17 of [52], it is not clear how a plurality of such surveys would be combined into one; but this is easily done in the L-valued case (for certain L) by using means or t-norms or other operators applied to the L-values recorded in the incidence matrix by each survey-taker. Such survey examples lead to formal L-contexts as defined in Definition 5.5. Additional examples of L-valued formal contexts are summarized in Example 5.4 using notions and terminology of data-mining [49]. These L-valued formal contexts are constructed from traditional formal contexts (G, M, R) and their associated Galois connections (H, K ), and such L-valued formal contexts confirm the relevance and applicability of the categories subsequently studied in this paper. Definition 5.1 (Stumme [49]). 1. A pattern of (G, M, R) is a subset of M. 2. Suppose G is finite and D is a pattern of (G, M, R). Then the support supp(D) of D is given by supp(D) =

|K (D)| . |G|

3. A pattern D is frequent with respect to  ∈ [0, 1] if supp(D) > , in which case  is denoted minsupp(D). It should be noted by Proposition 2.4 that supp(D) indicates the numerically largest proportion of G associated with D by (H, K ). Definition 5.2. The following operators from ℘(G)×℘(M) to a commutative, residuated lattice (L , ⱕ , ⊗) are defined as follows: 1. For L = [0, 1] with ⊗ = binary ∧: Rsupp1a (C, D) = supp(D) =  Rsupp1b (C, D) =

|K (D)| , |G|

supp(D), (C, D) a formal protoconcept, 0,

otherwise,

Rsupp2 (C, D) =

|C ∩ K (D)| , |G|

Rsupp3 (C, D) =

|K H (C) ∩ K (D)| . |G|

2. For L the usual co-topology on [0, 1] of closed subsets, equipped with the ordering ⊃ and ⊗ = binary ∪:  [supp(H (C)), supp(D)], (C, D) a formal preconcept, R Rsupp1 (C, D) = 0, otherwise. 3. For L = {[a, b] : a, b ∈ [0, 1]} ∪ {⭋}, equipped with the ordering ⊃ and ⊗ = binary ∩:  [supp(H (C)), supp(D)], (C, D) a formal preconcept, R Rsupp2 (C, D) = 0, otherwise.

68

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Proposition 5.3. The following hold: 1. Rsupp1a , Rsupp1b , and Rsupp3 coincide on formal protoconcepts. 2. Rsupp1a , Rsupp1b , Rsupp2 , and Rsupp3 coincide on formal concepts. Example 5.4. Each (℘(G), ℘(M), R) is a formal L-context, as defined in Definition 5.5 below, where R may be any of Rsupp1a , Rsupp1b , Rsupp2 , Rsupp3 , R Rsupp1 , and R Rsupp2 for the appropriate L. This section now gives the basic ideas for lattice-valued FCA, beginning with the definition of formal L-contexts and laying special stress on the lattice-valued counterparts of the Birkhoff operators given in Theorem 2.5. Definition 5.5. A formal L-context is an ordered triple (G, M, R), where G is the set of objects, M is the set of attributes, and R is an L-valued relation from G to M, i.e., R : G × M → L. For each g ∈ G, m ∈ M, R(g, m) is the degree to which object g has attribute m. We also write K for (G, M, R). The notion of a formal L-context can be found in [4–7]. There is also the notion of many-valued contexts given in [25]. Before taking up the issue of lattice-valued Birkhoff operators for formal L-contexts, it behooves us to examine the relationships between many-valued contexts and formal L-contexts. Let (G, M, W, I ) be a many-valued context in the sense of [25]. This means that I is a ternary relation on (G, M, W ), i.e., I ⊂ G × M × W , equipped with a certain “well-definedness” property: (g, m, w1 ), (g, m, w2 ) ∈ I ⇒ w1 = w2 . Choose  to be an element not in W and put L to be the poset formed by adjoining  to W, namely L := W ∪⭋ {}, the disjoint union of W with {}, equipped with the “flat” ordering  ⱕ  in L ⇔  =  or  = . Define R I : G × M → L by  w ∈ L , (g, m, w) ∈ I, R I (g, m) = , otherwise. Then the well-definedness property of I assures us that (G, M, R I ) is a formal L-context analogous to those defined in Definition 5.5, except that here L is a “flat” poset and not necessarily a commutative residuated semiquantale. This yields a well-defined correspondence I  R I , and this mapping is injective: if I  J as relations on G × M × W and, say, (g, m, w) ∈ I − J , then R I (g, m) = w (  )  w ∈ L , (g, m, w ) ∈ J  , otherwise = R J (g, m). Now let (G, M, R) be a formal L-context with L consistent. Put W = L − {⊥} with the inherited partial order, and put I R ⊂ G × M × W by (g, m, w) ∈ I R ⇔ w = R(g, m)  ⊥. This yields a well-defined correspondence R  I R , and this mapping is “essentially” injective, i.e., injective up to order-isomorphism as shown in the following argument: it suffices to consider the case when (G, M, R1 ) is a formal L 1 -context, (G, M, R2 ) is a formal L 2 -context, and I R1 = I R2 ; this forces L 1 − {⊥ L 1 } = L 2 − {⊥ L 2 };

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

69

and in this case the order-isomorphism between L 1 and L 2 sends one bottom element to the other and is otherwise the identity map. Finally, let (G, M, W, I ) be a many-valued context, choose L ∗ = ℘(W ), and put  {w} ∈ L ∗ , (g, m, w) ∈ I, R ∗I (g, m) = ⭋, otherwise. Then the well-definedness property of I assures us that (G, M, R ∗I ) is a formal L ∗ -context as defined in Definition 5.5—L ∗ as a traditional powerset is a commutative, strictly two-sided, residuated semiquantale (with ⊗ = binary ∩). This yields another well-defined correspondence I  R ∗I , and this mapping is injective: if I  J as relations on G × M × W and, say, (g, m, w) ∈ I − J , then R I (g, m) = {w} (  ⭋)  {w } ∈ L ∗ , (g, m, w ) ∈ J  ⭋, otherwise = R J (g, m). There are a number of conclusions to be drawn from the preceding paragraphs: • The correspondences I  R I and R  I R are “essentially” inverse to each other, i.e., yield a bijection “up to orderisomorphism” between many-valued contexts in the sense of [25] and the schema of formal L-contexts analogous to those given in Definition 5.5, providing L is “flat” and we ignore the standing assumptions in 5.5 that L be a commutative residuated semiquantale. To be more precise, it is the case that I R I = I and that R I R is bijective with R as sets of ordered triples and that the underlying lattices are order-isomorphic. • For a given many-valued context (G, M, W, I ), the two induced, many-valued relations R I and R ∗I , as sets of ordered triples, are bijective with each other via the correspondence  (g, m, {}),  ∈ W (g, m, ) ∈ R I  ∈ R ∗I . (g, m, ⭋),  =  This bijection lifts to say that the family {R I : (G, M, W, I ) is a many-valued context} is bijective with the family {R ∗I : (G, M, W, I ) is a many-valued context}. • The correspondence I  R ∗I indicates that the family of all many-valued contexts in the sense of [25] inject into the schema {formal L-contexts : L is a commutative residuated semiquantale} of all formal lattice-valued contexts provided by Definition 5.5, an injection which is not surjective since the standing assumption in 5.5 of a commutative residuated semiquantale is much broader than traditional powersets. It should be noted that Remark 9.4 below outlines a coherent, variable-basis, categorical setting for this schema of all formal lattice-valued contexts. Returning to the issue of lattice-valued Birkhoff operators, the following theorem, which appeared as part of Lemma 3 in Bˇelohlávek [3] in 1999, extends the Birkhoff operators of Theorem 2.5 to the L-valued setting. This theorem is foundational for the rest of the paper; and for the convenience of the reader, we include a proof of this theorem which is rather different than that given in [3] for Bˇelohlávek’s Lemma 3. Theorem 5.6 (L-Birkhoff operators). Let (G, M, R) be a formal L-context and let H : L G → L M , K : L M → L G be defined as follows:  (s(g) → R(g, m)), (H (s))(m) = g∈G

(K (t))(g) =



(t(m) → R(g, m)).

m∈M

Then (H, L G , L M , K ) is a Galois connection, where L G , L M are taken with the usual point-wise order.

70

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Proof. Clearly H, K are well-defined. As for antitonicity of H, K , we first appeal to the fact that ∀a, b, c ∈ L , a ⱕ b ⇒ (b → c) ⱕ (a → c), a fact which follows from the isotonicity of ⊗ and these observations which assume a ⱕ b: ∀e ∈ L , a ⊗ e ⱕ b ⊗ e, {e ∈ L : b ⊗ e ⱕ c} ⊂ {d ∈ L : a ⊗ d ⱕ c},   eⱕ d = (a → c). (b → c) = b⊗e ⱕ c

a⊗d ⱕ c

Now for the antitonicity of H, let s1 ⱕ s2 in L G and let m ∈ M. Then ∀g ∈ G, (s1 (g) → R(g, m)) ⱖ (s2 (g) → R(g, m)), so that (H (s1 ))(m) =



(s1 (g) → R(g, m)) ⱖ

g∈G



(s2 (g) → R(g, m)) = (H (s2 ))(m).

g∈G

The antitonicity of K is similarly proved. To complete the proof of the Galois connection, we need to show that H K ⱖ id L M ,

K H ⱖ id L G ,

and for this we need these two facts:



1. ∀  ⭋, ∀{a }∈ ⊂ L , ∀b ∈ L , (( ∈ a ) → b) ⱖ ∈ (a → b). 2. ∀a, b ∈ L , ((a → b) → b) ⱖ a. Fact (1) follows from the antitonicity of → justified above; and Fact (2) is given in [3] (p. 498) and its references. Now to show that H K ⱖ id L M , we distinguish two cases. Case 1: M = ⭋. In this case, L M is singleton, so that H K = id L M . Case 2: M  ⭋. In this case, we let t ∈ L M and m ∈ M, and note  ([K (t)](g) → R(g, m)). (H (K (t)))(m) = g∈G

Now, if G = ⭋, then (H (K (t)))(m) =  ⱖ t(m); and if G  ⭋, then (H (K (t)))(m) =





g∈G

ⱖ





(t(n) → R(g, n)) → R(g, m)

n∈M

 

([(t(n) → R(g, n))] → R(g, m)) ((1) above)

g∈G n∈M

ⱖ



([(t(m) → R(g, m))] → R(g, m))

g∈G

ⱖ



t(m) (Fact (2) above)

g∈G

= t(m). The proof that K H ⱖ id L G is similar.  We are now in a position to put some of the ideas from Section 2 into an explicitly lattice-valued framework as is done in [6]. Whenever “H” and “K” are used in the context of a formal L-context (G, M, R), it is to be assumed that we are referring to the H, K constructed in Theorem 5.2.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

71

Definition 5.7. Let K := (G, M, R) be a formal L-context. Then a formal L-concept of K is an ordered pair (a, b) ∈ L G × L M such that H (a) = b and K (b) = a. Definition 5.8. If (a, b), (c, d) are formal L-concepts of K := (G, M, R), then (a, b) ⱕ (c, d) if a ⱕ c or, equivalently, if d ⱕ b. Theorem 5.9. Let K := (G, M, R) be a formal L-context and (H, L G , L M , K ) be the Galois connection generated by K. Then the set of all formal L-concepts of K as ordered in Definition 5.8 is a complete lattice, and it is isomorphic to K → (L M ) and anti-isomorphic to H → (L G ). Theorem 5.9 leads to the following definition taken from [5,6,10]. Definition 5.10. The set of all formal L-concepts of K := (G, M, R) is called the L-concept lattice of K. The rest of this section records L-analogues of ideas and results in Section 2. Definition 5.11. Let K := (G, M, R) be a formal L-context, (H, L G , L M , K ) be the Galois connection generated by K, and (c, d) ∈ L G × L M . 1. The pair (c, d) is a formal L-preconcept of K if c ⱕ K (d) or, equivalently, d ⱕ H (c); and the collection of all formal L-preconcepts is ordered by (a, b)  (a , b ) if and only if a ⱕ a and b ⱕ b . 2. Precon(c, d) denotes the collection of all formal L-concepts (a, b) such that (c, d)  (a, b), together with the ordering of formal L-concepts. 3. The pair (c, d) is a formal L-protoconcept of K if (c, d) is a formal L-preconcept such that Precon(c, d) contains exactly one formal concept. Proposition 5.12. Let K := (G, M, R) be a formal L-context and (H, L G , L M , K ) be the Galois connection generated by K. I. For formal L-preconcept (c, d) ∈ L G × L M , Precon(c, d) is a complete lattice with smallest formal concept (K H (c), H (c)) and largest formal concept (K (d), H K (d)), and Precon(c, d) contains each formal concept (a, b) with K H (c) ⱕ a ⱕ K (d). II. The following are equivalent for a pair (c, d) ∈ L G × L M . (a) (b) (c) (d) (e) (f)

The pair (c, d) is a formal L-protoconcept of K. Precon(c, d) contains exactly one formal concept. K H (c) = K (d). H K (d) = H (c). The L-subsets c and d are respectively the elements of anti-isomorphic fibers of H and K . The pair (K (d), H (c)) is a formal L-concept of K.

Proposition 5.12(II)(f) justifies the term “L-protoconcept” introduced in Definition 5.11. 6. Category L-FCI for formal L-contexts As in Section 3, in order to facilitate mathematical developments in lattice-valued FCA, we construct a category whose objects are formal L-valued contexts and whose morphisms respect the associated formal L-protoconcepts and the structures associated with these L-protoconcepts. The motivations for the various conditions imposed on the morphisms of this category are the same as in Section 3 and are also formally necessary; cf. Theorems 3.3–3.4 and Theorems 6.2–6.3 and Section 9. Definition 6.1. The category L-FCI of L-formal context interchanges has ground category Set × Setop and comprises objects and morphisms as follows: 1. Objects are formal L-contexts of the form K := (G, M, R).

72

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

2. Morphisms are ( f, ) : K1 → K2 , where (H1 , L G 1 , L M1 , K 1 ) and (H2 , L G 2 , L M2 , K 2 ) are the Galois connections respectively generated by K1 := (G 1 , M1 , R1 ) and K2 := (G 2 , M2 , R2 ), and the following hold: (a) ( f, ) : (L G 1 , L M1 ) → (L G 2 , L M2 ) is a ground morphism, i.e., f : L G 1 → L G 2 and op : L M2 → L M1 are mappings. (b) Each of the following commutivities holds: H1 = op ◦ H2 ◦ f,

K 2 = f ◦ K 1 ◦ op .

Theorem 6.2. Let ( f, ) : K1 → K2 be an L-FCI morphism. The following hold: 1. Each of f and op respects the equivalence classes set up by the appropriate Galois connection. More precisely, if s1 , s2 ∈ L G 1 such that H1 (s1 ) = H1 (s2 ), then H2 ( f (s1 )) = H2 ( f (s2 )); and if t1 , t2 ∈ L M2 such that K 2 (t1 ) = K 2 (t2 ), then K 1 (op (t1 )) = K 1 (op (t2 )). 2. Each of f and op respects the fixed L-subsets set up by the appropriate Galois connection. More precisely, if s ∈ L G 1 such that K 1 H1 (s) = s, then K 2 H2 ( f (s)) = f (s); and if t ∈ L M2 such that H2 K 2 (t) = t, then H1 K 1 (op (t)) = op (t). Hence each of f and op respects leaf nodes. Further, ( f, ) “interchanges” formal concepts in the sense that ∀a ∈ L G 1 , d ∈ L M2 with a, d leaf nodes, (a, op (d)) is a formal L-concept if and only if ( f (a), d) is a formal L-concept. 3. Each of f and op is a bijection on leaf nodes. 4. ( f, ) “interchanges” formal L-protoconcepts in the sense that ∀c ∈ L G 1 , ∀d ∈ L M2 , it is the case that (c, op (d)) is a formal L-protoconcept of K1 if and only if ( f (c), d) is a formal L-protoconcept of K2 . 5. One of the maps f, op detects the ordering of leaf nodes if and only if the other map preserves the ordering of the leaf nodes. 6. f and op preserve orderings of leaf nodes if and only if ( f, ) “interchanges” formal L-preconcepts in this sense: ∀c ∈ L G 1 , ∀d ∈ L M2 , it is the case that (c, op (d)) is a formal L-preconcept of K1 if and only if ( f (c), d) is a formal L-preconcept of K2 . f → (L G 1 )

7. f and op preserve orderings of leaf nodes if and only if both f |(K op )→ (L M2 ) : (K 1 op )→ (L M2 ) → f → (L G 1 ) and 1 (op )→ (L M2 ) op |(H f )→ (L G 1 ) : (H2 f )→ (L G 1 ) → (op )→ (L M2 ) are order isomorphisms. 2

Proof. The proof from Definition 3.2 and the commutivities of Definition 6.1(2)(b) mirrors that of Theorem 3.3 from Definition 3.2 and the commutivities of Definition 3.1.  It should be pointed out that a lattice-valued counterpart L-GalConn to GalConn of Section 3 can be defined analogously to how L-FCI is a lattice-valued counterpart to FCI; and in this regard, the first four itemized bullets of Section 3 can be given L-valued counterparts which answer to Theorem 6.2(1–4). These four properties, as dealt with in Theorem 6.2(1–4) and Theorem 6.3, demonstrate that the morphisms definition for each L-FCI [L-GalConn] is a formal necessity for any categorial approach which uses “forward-and-backward” [“forward-and-forward”] L-powerset level maps and which “interchanges” [preserves] the formal analytic structure of formal L-contexts; cf. Section 9. Theorem 6.3 (Converse to Theorem 6.2(1–4)). Let K1 = (G 1 , M1 , R1 ), K2 = (G 2 , M2 , R2 ) be formal L-contexts. 1. If ( f, ) : (L G 1 , L M1 ) → (L G 2 , L M2 ) is a Set × Setop morphism satisfying the conditions of Theorem 6.2(1–4), then ( f, ) is a morphism in L-FCI. 2. If ( f, g) : (L G 1 , L M1 ) → (L G 2 , L M2 ) is a Set × Setop morphism satisfying the statements concerning L-GalConn in the above paragraph, then ( f, ) is a morphism in L-GalConn. Proof. The proof parallels that of Theorem 3.4(1–4) and is left to the reader.  The remarks made at the end of Section 3 regarding the richness of category FCI and its morphisms apply equally well to each L-FCI, including the distinctive relationships of the component maps of L-FCI morphisms to the orders of relevant powersets. We also point out the manner in which L-FCI is defined above is in anticipation of the next two

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

73

sections’ exploration of the relationships of L-FCI with both Chu2 and the category Chu L of L -Chu systems and L-Chu transforms. 7. Relationships between L-FCI, Chu L , Chu2 This section continues the work of relating contexts to systems by focusing on the relationships between the categories L-FCI, Chu L , Chu2 , relationships which extend to the L-valued case several ideas from Section 4. We begin with the definitions of Chu L and L-Chu extent spaces and L-Chu continuity in this lattice-valued system context, which definition, discussion, and subsequent proposition are from [38] and [11]: Definition 7.1. An L-Chu space or L-Chu system is a triple (X, A, ), where (X, A) ∈ Set×Setop ,  is an L-satisfaction relation from X to A, namely a mapping  : X × A → L, and A is the set of predicates associated with the objects of X. L-Chu transforms between L-Chu systems are ordered pairs ( f, ) : (X, A, 1 ) → (Y, B, 2 ), with ( f, ) ∈ Set × Setop , f : X → Y a set function, and  : A → B a Setop morphism, satisfying the L-adjointness property that for all x ∈ X and all b ∈ B, 2 ( f (x), b) = 1 (x, op (b)). The category Chu L comprises all L-Chu systems and L-Chu transforms, along with the compositions and identities inherited from Set × Setop , the latter being the ground category for Chu L . As in Section 4, we prefer the terminology “L-Chu system” over “L-Chu space” for these reasons: the system terminology emphasizes the connection with L-topological systems, L-measurable systems, systems arising from latticevalued predicate transformers, and other types of systems—see [11] and Section 8 below; the term “space” suggests a kind of (L-)topological setting, but it is easily seen that Chu L is not a topological category over its ground category; and the system terminology allows us to carefully distinguish between Chu L and the “L-Chu extent spaces” which it induces, spaces which are indeed part of a lattice-valued topological construct. As just mentioned, closely associated with L-Chu systems and L-Chu transforms are “L-Chu extent spaces” and “L-Chu-continuous” mappings. Given an L-Chu system (X, A, ), there is a mapping ext L : A → L X defined by the evaluation map ext L (a)(x) = (x, a) along with the associated L-Chu extent space (X, (ext L )→ (A)). Proposition 7.2. If ( f, ) : (X, A, 1 ) → (Y, B, 2 ) is an L-Chu transform, then f : (X, (ext L )→ (A)) → (Y, (ext L )→ (B)) has the property that ∀v ∈ (ext L )→ (B),

f L← (v) ∈ (ext L )→ (A),

where f L← : L Y → L X is the Zadeh preimage operator given by the evaluation f L← (v)(x) = v( f (x)). This proposition justifies saying that the map f is L-Chu continuous. The category L-ChuTop comprises L-Chu topological spaces (X, A) with A ⊂ L X , along with L-Chu continuous mappings f between the ground sets of L-spaces (X, A), (Y, B) such that ( f L← )→ (B) ⊂ A. It can be shown that L-ChuTop is topological over Set with respect to the forgetful functor; and the above discussion and Proposition construct the L-extent functor E xt L : Chu L → L-ChuTop. Links of these notions to L-infratopological systems and their associated extent spaces—which are L-infratopological spaces—scan be constructed as in Section 4. The next theorem insures a good supply of meaningful examples of objects and morphisms in L-FCI, and for this theorem it is important to have the subcategory Chu L em of Chu L comprising all objects of Chu L , together with all

74

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

morphisms ( f, ) which are epi-monos, namely that f is epi (surjective) in Set and  is mono in Setop (so that op is surjective in Set). Theorem 7.3. E em : Chu L em → L-FCI is a covariant functor, where E em (X, A, ) = (X, A, ), E em [( f, ) : (X, A, 1 ) → (Y, B, 2 )] = ( f L→ , ( L

op→ op

) ) : (X, A, 1 ) → (Y, B, 2 ),

:= (op )→ where f L← ,  L L are, respectively, the Zadeh image operators; and if L is consistent, then E em is an embedding. Further, Fem : Chu L em → L-FCI is a covariant functor, where op→

Fem (X, A, ) = (X, A, ), Fem [( f, ) : (X, A, 1 ) → (Y, B, 2 )] = ( f L⇒ , ( L

op⇒ op

and where f L⇒ = K 2 ◦ H2 ◦ f L→ and  L

op⇒

) ) : (X, A, 1 ) → (Y, B, 2 ), op→

= H1 ◦ K 1 ◦  L

.

Proof. We prove only the first assertion concerning E em : Chu L em → L-FCI; the assertion concerning Fem : Chu L em → L-FCI will necessarily follow. And concerning E em , the main task is to show that if ( f, ) : (X, A, ) → (Y, B, ) is epi-mono in Chu L , then ( f L→ , ( L

op→ op

) ) : (X, A, 1 ) → (Y, B, 2 )

is an L-FCI morphism. That ( f L→ , ( L )op ) satisfies Definition 6.1(2)(a) is immediate. Now we show that ( f L→ , op→ ( L )op ) satisfies 6.1(2)(b), and for this we need a lemma. op→

Lemma 7.3.1. In L ⎞ ⎛   ⎝ a ⎠ → c = (a → c). ∈

∈

Proof. Using the alternative definition of → from ⊗—namely a → c ⱖ d ⇔ a ⊗ d ⱕ c, we have ⎞ ⎛ ⎞ ⎛   ⎝ a ⎠ → c ⱖ d ⇔ ⎝ a ⎠ ⊗ d ⱕ c ∈

∈

⇔



(a ⊗ d) ⱕ c

∈

⇔ ∀  ∈ , a ⊗ d ⱕ c ⇔ ∀  ∈ , a → c ⱖ d  ⇔ (a → c) ⱖ d, ∈

from which the lemma follows. 

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

75

Resumption of proof of 7.3. Let s ∈ L X , m ∈ A. Then  op→  L (H2 ( f L→ (s)))(m) = H2 ( f L→ (s))(m ∗ ) op (m ∗ )=m

=











⎡

⎡



⎡





⎤ ( f L→ (s)(g ∗ ) → 2 (g ∗ , m ∗ ))⎦ ⎛⎛



⎝⎝

⎞

⎞⎤

s(g)⎠ → 2 (g , m )⎠⎦ ∗

f (g)=g ∗







g ∗ ∈Y

f (g)=g ∗





g ∗ ∈Y

f (g)=g ∗





∗

⎤

(s(g) → 2 (g ∗ , m ∗ ))⎦ (•)

g ∗ ∈Y f (g)=g ∗

⎣

op (m ∗ )=m

=



⎣

op (m ∗ )=m

=

⎡

g ∗ ∈Y

⎣

op (m ∗ )=m

=



⎣

op (m ∗ )=m

=

⎡

g ∗ ∈Y

⎣

op (m ∗ )=m

=



⎣

op (m ∗ )=m

=

⎡

⎤ (s(g) → 2 ( f (g), m ∗ ))⎦ ⎤ (s(g) → 1 (g, op (m ∗ )))⎦ ⎤ (s(g) → 1 (g, m))⎦

g ∗ ∈Y f (g)=g ∗

(s(g) → 1 (g, m)) (••)

g ∗ ∈Y f (g)=g ∗

=



(s(g) → 1 (g, m)) (• • •)

g∈X

= H1 (s)(m), where line (•) uses Lemma 7.3.1, line (••) results from the surjectivity of op and taking the join of a singleton subset, and in line (• • •) the surjectivity of f is used. Thus we have that op→

H1 =  L

◦ H2 ◦ f L→ .

The other commutivity K 2 = f L→ ◦ K 1 ◦  L

op→

.

follows similarly using first the surjectivity of f and then the surjectivity of op . op→ We now have that ( f L→ , ( L )op ) is an L-FCI morphism. Now E em preserves composition of morphisms since Zadeh image operators are closed under the composition of Set. Also, E em preserves identities since the Zadeh image operators of identity maps in Set are identity maps. To show that E em is an embedding, assume that L is consistent and suppose ( f, ), (g, ) : (X, A, ) → (Y, B, ) are Chu2em morphisms with ( f, )  (g, ). Suppose f  g. Then ∃x ∈ X such that f → ({x}) = { f (x)}  {g(x)} = g → ({x}). Now it is well-known that Zadeh image operators lift their traditional counterparts, i.e., ∀A ⊂ X f L→ ( A ) = f → (A) .

76

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Hence, f L→ ( {x} ) = f → {x}  g→ {x} = g → L ( {x} ). Thus f L→  g → hence ( f L→ , ( L )op )  (g → )op ). A similar proof shows that if   , then ( f L→ , L and op→ L , ( L op→ op ). Hence E , ( ) is faithful. Finally, E em trivially injects objects, so E em is an embedding. ( L )op )  (g → em L L This concludes the proof of the theorem.  op→

op→

We note from the counterexample given following Theorem 4.6 that Fem in Theorem 7.3 need not be an embedding even when L is consistent. Theorem 7.4. E I : L-FCI → Chu2 is a concrete functor defined as follows: E I (K := (G, M, R)) = (L G , L M , ), where  ⊂ L G × L M defined by c  d if and only if (c, d) is a formal L-protoconcept of K, and E I (( f, ) : K1 → K2 ) = [( f, ) : E I (K1 ) → E I (K2 )]. Further, E I is an embedding if L is consistent and  is an identity for ⊗. Proof. We first observe from Theorem 6.2(4) that E I ( f, ) ≡ ( f, ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ) is a Chu transform in Chu2 . By inspection, E I preserves composition and identities, and hence E I is functorial; and since E I is concrete, it is faithful. We now address the question, under the additional assumption that L is consistent and  is the identity of ⊗, whether E I injects objects, noting that since the correspondence of sets to L-powersets is injective, this question reduces to the following: given formal L-contexts (G, M, R1 ) and (G, M, R2 ) with R1  R2 , is it the case that 1  2 ? To that end, suppose (G, M, R1 ) and (G, M, R2 ) are formal L-contexts with R1 , R2 : G × M → L, and R1  R2 . Then ∃(g ∗ , m ∗ ) ∈ G × M with R1 (g ∗ , m ∗ )  R2 (g ∗ , m ∗ ). Let  := R1 (g ∗ , m ∗ ),  := R2 (g ∗ , m ∗ ). Now    implies that either  ⱕ 

or  ⱕ . So we may assume W.L.O.G. that  ⱕ . This assumption, together with the infinite distributivity of ⊗ over and the definition of the residuation →, forces  →  < . With these preliminary remarks in place, we are ready to construct c ∈ L G and d ∈ L M such that c 1 d and c 2 d. Put c = {g∗ } , d = H1 (c). Then it is the case immediately that K 1 H1 (c) = K 1 (d), so that (c, d) is an L-protoconcept of (G, M, R1 ), and hence c 1 d.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

77

It is now necessary to make a series of derivations. First, for m ∈ M d(m) = H1 (c)(m)  (c(g) → R1 (g, m)) = g∈G

= c(g ∗ ) → R1 (g ∗ , m) =  → R1 (g ∗ , m) = R1 (g ∗ , m). Second, again for m ∈ M, and by a similar derivation, it follows: H2 (c)(m) = R2 (g ∗ , m). Third, using the derivation of d, we have  K 2 (d)(g ∗ ) = (d(m) → R2 (g ∗ , m)) m∈M



=

(R1 (g ∗ , m) → R2 (g ∗ , m))

m∈M

ⱕ R1 (g ∗ , m ∗ ) → R2 (g ∗ , m ∗ ) = → < . And, fourth, using the derivation of H2 (c), we have  K 2 H2 (c)(g ∗ ) = (H2 (c)(m) → R2 (g ∗ , m)) m∈M

=



(R2 (g ∗ , m) → R2 (g ∗ , m))

m∈M

=





m∈M

= . It follows that K 2 H2 (c)  K 2 (d), so that (c, d) is not an L-protoconcept of (G, M, R2 ), and hence c 2 d. Thus E I (G, M, R1 ) = (L G , L M , 1 )  (L G , L M , 2 ) = E I (G, M, R2 ), and E I is seen to inject objects. Since E I is faithful and injects objects, it is a functorial embedding, concluding the proof.  Corollary 7.5. Let L be consistent. Then L-FCI maps into Chu L via a concrete functor; and if, further,  is an identity for ⊗, then L-FCI functorially embeds into Chu L .

78

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Proof. Using the previous theorem, the needed functors are given by E I followed by the functorial embedding Chu2 : Chu2 → Chu L , where the latter is given by Chu2 (X, A, ) = (X, A,  L ), Chu2 ( f, ) = ( f, ), where



 L : X × A → L by  L (x, a) =

 , x  a, ⊥,

x  a.



To return to the end of Section 4, Theorem 4.7 now follows from Corollary 7.5 by choosing L = 2, noting that Chu2 for such L is an isomorphism. 8. Positioning (L)-TopSys within (L)-Chu systems using (L )-FCI This section continues to develop the relationships between formal contexts and systems by indicating how the former can be used as a tool to give additional insights into the relationships between (L)-topological and (L)-Chu systems. These relationships show (L)-Chu systems to be an appropriately large framework. But these relationships also give additional justification for (L)-Chu systems as a common rubric for topological systems, measurable systems, or any other kind of similarly structured systems, as well as formal contexts, since these structured systems simply cannot accommodate formal contexts reinterpreted via I and E I as systems. In order to state the questions which motivate this section, we need the following inventory of categories of systems defined using the objects and morphisms of Chu2 and Chu L , justifying examples for which categories are given in [11,13]: • TopSys [53] comprises those Chu systems (X, A, ), called topological systems, for which A is equipped with the structure of a locale and  satisfies the join and finite meet interchange laws  ∀x ∈ X, {a }∈ ⊂ A, x  a ⇔ ∃0 ∈ , x  a0 , ∈

∀x ∈ X, {a }∈ ⊂ A with  finite, x 



a ⇔ ∀ ∈ , x  a ,

∈

together with all Chu transforms ( f, ) between topological systems for which op is a frame morphism. A supratopological system is a Chu system for which A is equipped with the structure of a complete lattice and  satisfies the join interchange law; a finitely supratopological system is a Chu system for which A is equipped with the structure of a join-semilattice and  satisfies a join interchange law having only finite indexing sets ; an infratopological system is a Chu system for which A is equipped with the structure of a complete lattice and  satisfies a meet interchange law in which the indexing sets  are arbitrary; a finitely infratopological system is a Chu system for which A is equipped with the structure of a meet-semilattice and  satisfies the finite meet interchange law; and an Alexandrov topological system is a Chu system for which A is equipped with the structure of a complete lattice and  satisfies join and meet interchange laws with arbitrary indexing sets. • MeasSys [11,13] comprises those Chu systems (X, A, ), called measurable systems, for which A is equipped with the structure of a -Boolean algebra,  satisfies the join and meet interchange laws with countably infinite indexing sets, and  satisfies the negation interchange law ∀x ∈ X, a ∈ A, x  ¬a ⇔ x  a, together with all Chu transforms ( f, ) between measureable systems for which op is a -Boolean morphism (preserving all -joins and negations); and a Boolean system (X, A, ) is defined analogously except that A is equipped with the structure of a Boolean algebra and all indexing sets for the join and meet interchange laws are finite.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

79

• L-TopSys [11,13,46,47], with L a frame, comprises those L-Chu systems (X, A, ), called L-topological systems, for which A is equipped with the structure of a locale and  satisfies the L-join and finite L-meet interchange laws ⎞ ⎛   ∀x ∈ X, ∀{a }∈ ⊂ A,  ⎝x, a ⎠ = (x, a ), ∈

⎛

∈

∀x ∈ X, ∀{a }∈ ⊂ A with  finite,  ⎝x,



⎞ a ⎠ =

∈



(x, a ),

∈

together with all L-Chu transforms ( f, ) between L-topological systems for which op is a frame morphism. The notions of L-supratopological, L-finitely supratopological system, L-infratopological system, L-finitely infratopological system, L-Alexandrov topological system are defined for appropriate conditions on L by modifying the definition of an L-topological system in a manner exactly analogous to that in which the correspondingly named notions above modify the definition of a topological system. • L-MeasSys [11,13] comprises those L-interchange systems (X, A, ), called measurable systems, for which L and A are -Boolean algebras,  satisfies the L-join and L-meet interchange laws with countably infinite indexing sets, and  satisfies the L-negation interchange law ∀x ∈ X, ∀a ∈ A, (x, ¬a) = ¬(x, a), together with all Chu transforms ( f, ) between L-measureable systems for which op is a -Boolean morphism; and an L-Boolean system (X, A, ) is defined analogously for appropriate L to how Boolean systems are defined above. As pointed out in [13], each of the categories given above in the first two [second two] bullets has an obvious forgetful functor from that category to Chu2 [Chu L ] which forgets the structure on the set A of predicates and any requirements on the satisfaction relation  associated with that structure on A as well as the requirement that opposites of second components of morphisms preserves A’s structure. For example, the functor V : TopSys → Chu2 is defined by V (X, A, ) = (X, A, ), V ( f, ) = ( f, ). To achieve the claims of the opening paragraph of this section, we primarily focus on the relationships between TopSys and I and E I , a focus motivated in part by the following two questions: Question 8.1. Does the embedding I : FCI → Chu2 produce topological systems or other Chu systems with structure (see Theorem 4.4)? Question 8.2. Does the embedding Chu2 ◦ E I : L-FCI → Chu2 → Chu L produce L-topological systems or other L-Chu systems with structure, where L is a complete, commutative, residuated, consistent lattice with  as the identity for the tensor product ⊗ (see Theorem 7.4, Corollary 7.5)? The functor of the following lemma is needed in this section, including the proof of Proposition 8.5 below. And this lemma needs the well-known notion of the support of a fuzzy subset: given a ∈ L X , the support of a, denoted supp(a), is {x ∈ X : a(x) > ⊥}. Lemma 8.3. For (G, M, R) ∈ |FCI|, put FCI (G, M, R) = (G, M, R ), where R : G × M → L by R (g, m) =  ⇔ (g, m) ∈ R. And for ( f, ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ), put FCI ( f, ) = ( f, op ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ),

80

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

where f (s) := f (supp(s)) , op (t) := op (supp(t)). The following hold: 1. If L is a complete, consistent chain with ⊗ = ∧, then FCI : FCI → L-FCI is an embedding. 2. If L = 2, then FCI : FCI → 2-FCI is an isomorphism. Proof. It is needful to first establish the relationship between the Birkhoff operators (H, K ) of a formal context (G, M, R) and the 2-Birkhoff operators (H , K ) of the 2-context (G, M, R ). Letting S ⊂ G and m ∈ M, it follows that  H ( S )(m) = ( S (g) → R (g, m)) g∈G

=



( S (g) → R (g, m))

g∈S

=



R (g, m)

g∈S

=



{m∈M:(g, ˆ m)∈R ˆ } (m)

g∈S

= g∈S {m∈M:(g, (m) ˆ m)∈R} ˆ = g∈S H ({g}) (m) = H (S) (m). so that H = H ; and similarly, K = K . For the proof of (1), let L be a complete chain with ⊗ = ∧. We first note that ∀a ∈ L   , a = ⊥, a→⊥= ⊥, a > ⊥. Now for to be a functor, it needs to be shown that Definition 6.1(2)(b) is satisfied. To that end, let ( f, ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ). To establish the first required identity H1 = op ◦ H2 ◦ f of 6.1(2)(b), let s ∈ L G 1 . Noting that 2 ⊂ L by consistency of L, we first confirm this identity for the special case when s ∈ 2G 1 , namely the case when there is S ⊂ G 1 such that s = S . Then, using the preceding paragraph, the fact that Zadeh image operators lift the traditional image operators via characteristics, and the fact that S = supp(s), it follows that



( op ◦ H2 ◦ f )(s) = op H2 f ( S )

= op H2 ( f (S) ) = op ( H2 ( f (S)) ) = op (H2 ( f (S))) = H1 (S)

= H1 S

= H1 (s),

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

81



so that H1 = op ◦ H2 ◦ f on 2G 1 . Now for the general case when s ∈ L G 1 , let m ∈ M1 . Then, on one hand  H1 (s)(m) = (s(g) → R1 (g, m)) g∈G 1



=

(s(g) → R1 (g, m))

g∈supp(s)



=



 s(g) →

g∈supp(s)



=

 , (g, m) ∈ R1 ⊥,



(g, m) ∈ / R1

(s(g) → ⊥)

g∈supp(s),(g,m)∈R / 1

 =

⊥,

∃ g ∈ supp(s), (g, m) ∈ / R1 ,

 , otherwise,

where the next to last line uses that L is a consistent complete chain with ⊗ = ∧. Now, on the other hand, using the special crisp case above with S replaced by supp(s), it follows that



op (H2 (( f )(s)))(m) = op (H2 ( f (supp(s))))(m) = op ( H2 ( f (supp(s))) )(m) = op [supp( H

2 ( f (supp(s)))

)] (m)

= op (H2 ( f (supp(s)))) (m) = H1 (supp(s)) (m)   , m ∈ H1 (supp(s)) = ⊥, m ∈ / H1 (supp(s))    , m ∈ {{m ∗ ∈ M1 : (g, m ∗ ) ∈ R1 } : g ∈ supp(s)} =  ⊥, m ∈ / {{m ∗ ∈ M1 : (g, m ∗ ) ∈ R1 } : g ∈ supp(s)}  ⊥, ∃ g ∈ supp(s), (g, m) ∈ / R1 =  , otherwise.





It is concluded that H1 = op ◦ H2 ◦ f on L G 1 . A dual and similar proof establishes the second identity K 2 = f ◦ K 1 ◦ op of Definition 6.1(2)(b). And so it follows that is a functor. But the consistency of L implies that is injective on morphisms, so is an embedding, concluding the proof of (1). For the proof of (2), we note the proof that is a functor is precisely the special case needed and proved above for (1). Now the relationships s ∈ 2G  S := {g ∈ G : s(g) = }, t ∈ 2 M  T := {m ∈ M : t(m) = }, may be used in the expected way to define and verify the inverse functor −1 FCI : 2-FCI → FCI. Thus, is an isomorphism, concluding the proof of the lemma.  Definition 8.4. A formal context (G, M, R) satisfies the one leaf condition if the associated Galois connection (H, K ) between ℘(G) and ℘(M) has exactly one (nonempty) leaf in ℘(G) (and, equivalently, in ℘(M)); and an L-formal context (G, M, R), for L a complete, commutative, residuated lattice, satisfies the one-leaf condition if the associated Galois connection (H, K ) between L G and L M has exactly one (nonempty) leaf in L G (and, equivalently, in L M ).

82

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

In each of the following results, (G, M, R) is a formal context with Galois connection (H, K ) and (℘(G), ℘(M), ) is the Chu system I(G, M, R); and in the L-valued case, (G, M, R) is an L-formal context with Galois connection (H, K ) and (L G , L M , ) is the Chu system E I (G, M, R). The next proposition restates the previous definition. Proposition 8.5. Let  be an identity for ⊗. Then a formal context (G, M, R) satisfies the one-leaf condition if and only if R is the universal relation, i.e., R = G × M; and an L-formal context (G, M, R) satisfies the one-leaf condition if and only if R is the L-universal relation, i.e., R =  on G × M, where ∀(g, m) ∈ G × M, (g, m) = . Proof. We prove only the L-valued case, for then the traditional case follows from Lemma 8.3 for L = 2; and we begin with some preliminary remarks to set up the proofs of necessity and sufficiency. Recall from Proposition 2.4(3) that, for R : G × M → L with Galois connection (H, K ), each leaf in L G is a fiber of the form H ← {t} for some t ∈ H → (L G ). This means that (G, M, R) satisfies the one-leaf condition if and only if ∀t ∈ H → (L G ), H ← {t} = L G , i.e., ∀t ∈ H → (L G ), ∀s1 , s2 ∈ L G , H (s1 ) = t = H (s2 ). Hence, (G, M, R) satisfies the one-leaf condition if and only if H is constant on L G ; and it is equivalent to say that K is constant on L M . Since it is the case for all Galois connections (I, J ) between L G and L M that I (⊥) =  and J (⊥) = , it follows that (G, M, R) satisfies the one-leaf condition if and only if ∀s ∈ L G , H (s) =  if and only if ∀t ∈ L M , K (t) = . We also note for the sequel that  being the identity of ⊗ implies that  → b = b universally holds in L. Now to prove necessity, assume that (G, M, R) satisfies the one-leaf condition, and let (g, m) ∈ G × M. Then, by the constancy of H , it follows: {m} ⱕ H K ( {m} ) = H ( {g} ). And it therefore follows that  = {m} (m) ⱕ H ( {g} )(m)  = ( {g} (g ∗ ) → R(g ∗ , m)) g ∗ ∈G

=  → R(g, m) = R(g, m). Hence R =  on G × M. For sufficiency, assume R =  on G × M and let s ∈ L G . Then   (s(g) → R(g, m)) = (s(g) → ) = . H (s)(m) = g∈G

g∈G

It follows from our set up above that (G, M, R) has the one-leaf condition.  Lemma 8.6. If either of the following hold, then (G, M, R) satisfies the one-leaf condition: 1. (℘(G), ℘(M), ) is a finitely supratopological system. 2. (℘(G), ℘(M), ) is a finitely infratopological system. Proof. Ad (1) Suppose (℘(G), ℘(M), ) is a finitely supratopological system, and let D1 , D2 ∈ ℘(M). Choose C1 = K (D1 ∪ D2 ). Then H (C1 ) = H K (D1 ∪ D2 ),

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

83

and hence C 1  D1 ∪ D2 . Since  satisfies the (finite) join-interchange law, it follows: C1  D1 or C1 D2 . Without loss of generality, we have C1  D1 . Then choosing C2 = K (D2 ), we have H (C2 ) = H K (D2 ), in which case C 2  D2 and C 2  D1 ∪ D2 , the latter statement following from the fact that  satisfies the (finite) join-interchange law. To sum up, the following four statements are now true: C 1  D1 , C 1  D1 ∪ D2 , C 2  D1 ∪ D2 , C 2  D2 . First, we infer that H (C1 ) is in the same leaf of ℘(M) as each of D1 and D1 ∪ D2 , forcing D1 and D1 ∪ D2 into the same leaf; and, second, we infer that H (C2 ) is in the same leaf of ℘(M) as each of D1 ∪ D2 and D2 , forcing D1 ∪ D2 and D2 into the same leaf. Hence D1 and D2 are in the same leaf of ℘(M). Since D1 , D2 are arbitrary, we conclude there is only one leaf in ℘(M). But the leaves of ℘(G) and ℘(M) are in a bijection, so ℘(G) has only one leaf. It follows that (G, M, R) has the one-leaf condition. Ad (2) The proof is somewhat analogous to that for (1). Suppose (℘(G), ℘(M), ) is a finitely infratopological system, and let D1 , D2 ∈ ℘(M). Choose C1 = K (D1 ∩ D2 ). Then H (C1 ) = H K (D1 ∩ D2 ), and hence C 1  D1 ∩ D2 . Since  satisfies the (finite) meet-interchange law, it follows C1  D1 and C1  D2 . Hence, each of (C1 , D1 ) and (C1 , D2 ) is a protoconcept in (G, M, R); and by Theorem 2.12(5), we may write K (D1 ) = K H (C1 ) = K (D2 ), forcing D1 , D2 into the same leaf of ℘(M). Now by the same argument used for (1), (G, M, R) has the one-leaf condition.  Lemma 8.7. Let (G, M, R) satisfy the one-leaf condition. Then the following hold: 1. (℘(G), ℘(M), ) is an infratopological system in which each C  ⭋.

84

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

2. (℘(G), ℘(M), ∗ ) is a supratopological system, where ∀C ∈ ℘(G), ∀D ∈ ℘(M) C ∗ D ⇔ D  ⭋ and C  D, or, equivalently C ∗ D ⇔ D  ⭋ and H (C) = H K (D). Proof. Recall that C  D if and only if (C, D) is a protoconcept in (G, M, R), i.e., if and only if K H (C) = K (D) by Theorem 2.12(5). Now assume that (G, M, R) has the one leaf condition. Then by the first paragraph of the proof of Proposition 8.5, we have that K is constant on ℘(M). It follows that ∀(C, D), K H (C) = K (D), so that C  D. Hence,  is the universal relation on ℘(G) × ℘(M). Consequently, the arbitrary meet-interchange law follows immediately as does the fact that each C  ⭋, justifying (1). Finally, statement (2) also follows, noting that the empty indexing set case for the arbitrary join-interchange law is handled by the stipulation that no C satisfies ⭋.  What are the properties of the correspondence (G, M, R)  (℘(G), ℘(M), ∗ ), given in Lemma 8.7(2)? Proposition 8.8. Let FCI⭋ denote the subcategory of FCI of all objects and morphisms ( f, ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ) such that ∀D2 ∈ ℘(M2 ), D2 = ⭋ if and only if op (D2 ) = ⭋. Then I⭋ : FCI⭋ → Chu2 is an embedding, where I⭋ (G, M, R) = (℘(G), ℘(M), ∗ ), I⭋ ( f, ) = ( f, ). Proof. I⭋ is an embedding since it is a restriction of the embedding I (Theorem 4.7) to the subcategory FCI⭋ .  Theorem 8.9. Let (G, M, R) be a formal context with Galois connection (H, K ) and (℘(G), ℘(M), ) be the Chu system I(G, M, R). Then (℘(G), ℘(M), ) cannot be any of the following: 1. 2. 3. 4. 5.

a finitely infratopological system in which some C does not satisfy ⭋; a topological system; a Boolean system; a measurable system; an Alexandrov topological system.

Proof. Assuming that (℘(G), ℘(M), ) is a finitely infratopological system in which no C satisfies ⭋ insures that (G, M, R) has the one-leaf condition by Lemma 8.6(2), in which case we have a contradiction by Lemma 8.7(1), verifying (1). Statement (2) follows from (1) using both the finite meet-interchange law and the fact that the joininterchange law insures that no C satisfies ⭋; and each of statements (3)–(5) follows in a similar way from (1).  Theorem 8.10. Let (G, M, R) be a formal context with Galois connection (H, K ) and (℘(G), ℘(M), ) be the Chu system I(G, M, R). Then the following hold: 1. (℘(G), ℘(M), ) is an infratopological system ⇔ (G, M, R) has the one-leaf condition ⇔ R = G × M. 2. (℘(G), ℘(M), ∗ ) is a supratopological system if (G, M, R) has the one-leaf condition, and the latter holds ⇔ R = G × M. Proof. For the first parts of each of (1) and (2), conjoin Lemmas 8.6(2) and 8.7(1) for (1), and use Lemma 8.7(2) for (2). Note the right hand equivalences in both (1) and (2) come immediately from Proposition 8.5.  Definition 8.11. A Chu system of the form (X, A, ), with A a complete residuated lattice, is a tensor Chu system if the following tensor interchange law holds: ∀x ∈ X, ∀a, b ∈ A, x  a ⊗ b ⇔ x  a and x  b.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

85

Lemma 8.12. If any of the following hold, then the L-context (G, M, R) satisfies the one-leaf condition: 1. (L G , L M , ) is a finitely supratopological system. 2. (L G , L M , ) is a finitely infratopological system. 3. (L G , L M , ) is a tensor Chu system. Proof. The proof of (1) follows that of Lemma 8.6(1) with the binary ∨ in lieu of the binary union ∪; the proof of (2) follows that of Lemma 8.6(2) with the binary ∧ in lieu of the binary intersection ∩; and the proof of (3) follows that of Lemma 8.6(2) with the ⊗ in lieu of the binary intersection ∩.  Lemma 8.13. Let L-context (G, M, R) satisfy the one-leaf condition. Then the following hold: 1. (L G , L M , ) is an infratopological system in which each c  ⊥. 2. (L G , L M , ∗ ) is a supratopological system, where ∀c ∈ L G , ∀d ∈ L M , c ∗ d ⇔ [d  ⊥ and H (c) = H K (d)]. 3. (L G , L M , ) is a tensor Chu system in which each c  ⊥. Proof. The proofs of (1) and (2) are respectively analogous to those of (1) and (2) of Lemma 8.7 à la Lemma 8.12. The proof of (3) follows that of Lemma 8.7(1) with ⊗ in lieu of the binary intersection ∩. To see that c  ⊥, it must be shown that K H (c) = K (⊥) (). Now the one-leaf condition forces the left-hand side of () to be . And to get the right-hand side of () to be , we compute for each g ∈ G:  K (⊥)(g) = (⊥(m) → R(g, m)) m∈M

=



(⊥ → R(g, m))

m∈M

=  (). The equality in () needs the property that ∀b ∈ L, ⊥ → b =  () and the justification   ⊥→b= c= c= ⊥⊗c ⱕ b

⊥ⱕb

of () needs that ⊥ is the (left-hand) annihilator for ⊗, a fact which follows from ⊗ distributing across arbitrary joins (and hence the empty join) on both sides.  What are the properties of the correspondence (G, M, R)  (L G , L M , ∗ ) given in Lemma 8.13(2)? Proposition 8.14. Let L-FCI⊥ denote the subcategory of L-FCI of all objects and morphisms ( f, ) : (G 1 , M1 , R1 ) → (G 2 , M2 , R2 ) such that ∀d2 ∈ L M2 , d2 = ⊥ if and only if op (d2 ) = ⊥. Then E I⊥ : L-FCI⊥ → Chu2 is an embedding, where E I⊥ (G, M, R) = (L G , L M , ∗ ),

E I⊥ ( f, ) = ( f, ).

86

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

Proof. E I⊥ is an embedding since it is a restriction of the embedding E I (Theorem 7.4) to the subcategory L-FCI⊥ .  Theorem 8.15. Let (G, M, R) be an L-formal context with Galois connection (H, K ) and (L G , L M , ) be the Chu system E I (G, M, R). Then (L G , L M , ) cannot be any of the following: 1. 2. 3. 4. 5. 6. 7. 8.

a finitely infratopological system in which some c does not satisfy ⊥; a tensor Chu system in which some c does not satisfy ⊥; a topological system; a tensor topological system; a Boolean system; a tensor Boolean system; a measurable system; an Alexandrov topological system.

Proof. The proof from Lemmas 8.12 and 8.13 is analogous to the proof of Theorem 8.9 from Lemmas 8.6 and 8.7.  Corollary 8.16. Let L be consistent, let (G, M, R) be an L-formal context with Galois connection (H, K ) and let (L G , L M , ) be the Chu system ( Chu2 ◦ E I )(G, M, R). Then (L G , L M , ) cannot be any of the following: 1. 2. 3. 4. 5. 6. 7. 8.

a finitely L-infratopological system in which some c does not satisfy ⊥; an L-tensor Chu system in which some c does not satisfy ⊥; an L-topological system; an L-tensor topological system; an L-Boolean system; an L-tensor Boolean system; an L-measurable system; an Alexandrov L-topological system.

Theorem 8.17. Let (G, M, R) be an L-formal context with Galois connection (H, K ) and (L G , L M , ) be the Chu system E I (G, M, R). Then the following hold: 1. (L G , L M , ) is an infratopological system ⇔ (G, M, R) has the one-leaf condition ⇔ R = . 2. (L G , L M , ) is a tensor Chu system ⇔ (G, M, R) has the one-leaf condition ⇔ R = . 3. (L G , L M , ∗ ) is a supratopological system if (G, M, R) has the one-leaf condition, and the latter holds ⇔ R = . Proof. Combine Lemma 8.12, Lemma 8.13, and Proposition 8.5.  Corollary 8.18. Let L be consistent, let (G, M, R) be an L-formal context with Galois connection (H, K ) and let (L G , L M , ) be the Chu system ( Chu2 ◦ E I )(G, M, R). Then the following hold: 1. (L G , L M , ) is an L-infratopological system ⇔ (G, M, R) has the one-leaf condition ⇔ R = . 2. (L G , L M , ) is a L-tensor Chu system ⇔ (G, M, R) has the one-leaf condition ⇔ R = . 3. (L G , L M , ∗ ) is a supratopological system if (G, M, R) has the one-leaf condition, and the latter holds ⇔ R = . Remark 8.19 (Summary of Section 8). 1. Focusing on FCI and TopSys as mapped into Chu2 by I and the forgetful functor V, respectively, where the latter maps topological systems to underlying Chu systems and continuous functions to underlying Chu transforms, this section shows that FCI and TopSys are disjoint from each other in Chu2 ; more precisely |E I→ (FCI)| ∩ |V → (TopSys)| = ⭋. This is also the case for any of the other structured Chu systems considered in this paper. Such disjointness shows the necessity of Chu2 in order to have a systems rubric which includes formal contexts and FCA along with Chu

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

87

systems stemming from predicate semantics and predicate transformers. Such conclusions can be repeated for the lattice-valued case if L is consistent. Specifically, for L-TopSys, |( Chu2 ◦ E I )→ (L-FCI)| ∩ |VL→ (L-TopSys)| = ⭋ in Chu L , where VL : L-TopSys → Chu L maps L-topological systems to underlying L-Chu systems and L-continuous functions to underlying L-Chu transforms; and it is similarly the case for other L-structured systems considered in this paper. So again, this shows the necessity of Chu L in order to have an L-systems rubric which includes L-formal contexts and L-FCA along with L-systems motivated by predicate semantics and predicate transformers. 2. It should be pointed out that the disjointness of (L-)FCI and (L-)TopSys as embedded within (L-) Chu systems reflects the definition of (L-)FCI, including its morphisms. But as pointed out in Section 1 above and Section 9 below, the morphisms of (L-)FCI are necessitated by compatibility with Chu transforms and, most importantly, an unrestricted preservation of the Galois connections associated with formal contexts, the only structure which formal contexts have. Hence, this disjointness is a consequence primarily of morphisms unconditionally preserving the structures of formal contexts. 9. Summary and future directions We summarize several themes of this paper and indicate possible directions for future research via a series of remarks. Remark 9.1. This paper links together topological systems arising from domain theory, Chu systems arising from predicate transformers and non-deterministic programming, formal contexts arising from a variety of applications and potential applications, lattice-valued extensions of such notions, and applications-oriented examples motivating many of these notions. Central to this linkage is the construction of FCI and L-FCI as categorical models of FCA and their morphisms. This construction is guided by several key insights: 1. A formal (L-)context is a(n) (L-)relation with a certain interpretation; and this relation, apart from its being welldefined as a subset (crisp or lattice-valued), has no structure. 2. The structure of a formal (L-)context, crisp or lattice-valued, is invested solely in the overlying structure of the Galois connection of its (L-)Birkhoff operators at the (L-)powerset level; and, we note in the crisp case, the family of relations between two sets is in a bijection with the family of antitone Galois connections between the powersets of the two sets. This overlying (L-)structure builds all the signature notions—(L-)concept, (L-)preconcept, (L-)protoconcept—which ab initio justify interpretating the underlying relation as a “formal (L-)context”—without such notions there is no reason to call the relation a formal (L-)context. 3. The authors view it as essential that a categorical context for (L-)relations called formal (L-)contexts should have morphisms that interchange/preserve the (L-)Birkhoff operators undergirding the notions of FCA; and such morphisms with forward–backward [forward–forward] components must be defined as in (L-)FCI [(L-)GalConn]—this necessity of definition is a consequence of Theorems 3.3–3.4 and Theorems 6.2–6.3. This results in apparently new approaches to categorical modeling of FCA and its lattice-valued extensions formally given in Definitions 3.1 and 6.1 and Remark 9.4. 4. It was a goal of this paper to link FCA to categorical frameworks for various notions of systems (topological, Chu), and this basically forces the choice of morphisms to be as in FCI and its lattice-valued siblings, namely the morphisms must display the forward-and-backward “adjointness” behavior consistent with morphisms associated with known categories of systems. However, it is a future research goal to study GalConn and lattice-valued siblings as alternative categorical foundations of FCA. Remark 9.2. As a trailer to Remark 9.1, it should be noted that if one takes the morphisms ( f, ) between formal contexts (G 1 , M1 , R1 ), (G 2 , M2 , R2 ) proffered by R. Ganter and Wille [25] (g, m) ∈ R1 ⇒ ( f (g), (m)) ∈ R2 , then the structures provided by the overlying Galois connections are almost never preserved by the image operators of the Ganter-Wille morphisms; and this is the case for a number of reasons, chiefly because the requirement in the above

88

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

display is not a biconditional. Even if the Ganter-Wille morphisms are significantly restricted to satisfy the biconditional requirement (g, m) ∈ R1 ⇔ ( f (g), (m)) ∈ R2 , then the Galois connections are necessarily preserved by the associated image operators only when both morphism components are surjective, an additionally severe restriction which in many easily constructed examples is not satisfied. Similar irresolvable problems occur when forward-and-backward modifications to the Ganter-Wille morphisms are made so that their associated image operators have the same directionality as is standard for systems morphism components: in this case the associated image operators do not generally interchange the overlying Galois connections as defined by Definition 3.1 except for those morphisms restricted to satisfy the biconditional adjointness condition and for which both concrete components are surjective. Finally, one of the referees suggested another variation of these morphisms ( f, ) with f arbitrary and op a bijection; but not only is the bijectivity of the second component a severe restriction, but the associated image operators will not generally interchange overlying Galois connections unless the first component is additionally a surjection—again, there are many easily constructed examples in which these conditions are not satisfied. To summarize: the Ganter-Wille morphisms, as shown by our analysis, do not interchange/preserve the structure of formal contexts; and this remains the case with many modifications of the Ganter-Wille morphisms, some of them rather severe; and thus the Ganter-Wille morphisms are suitable for a variation of the category Rel, but are not suitable as a categorical foundation for FCA if the structures studied in FCA are to be interchanged/preserved by the morphisms of that foundation. Remark 9.3. It should be noted that FCI and L-FCI may be regarded as implicitly “variable-basis” categories in the following sense: Chu systems provide a general systems rubric that includes topological systems and Chu systems arising from predicate transformers [13]; and it is known [14] that TopSys is an implicitly variable-basis category because, in part, it embeds in two fundamental and concrete ways into Loc-Top, the latter belonging to a schema of explicitly variable-basis categories for topology and fuzzy topology—see [41] and its references. These embeddings reveal that the locale of predicates of a topological system is really a lattice of memberships at the base of a lattice-valued powerset which varies from system/space to system/space in TopSys, showing the latter to be implicitly a variable-basis category. Remark 9.4. As a trailer to Remark 9.3, it would be interesting to consider an explicitly variable-basis approach to formal contexts, an approach in which the commutative residuated lattice-theoretic base L varies from context to context, and which, in light of Remark 9.3, would give us a kind of “doubly-variable-basis” approach to formal contexts. We now sketch such an approach. Let AbRSQuant denote the category of commutative residuated semiquantales with morphisms those mappings which preserve arbitrary joins and tensor products, and let AbRSQuantop denote its dual category. The category AbRSQuantop -FCI has ground category Set × Setop × AbRSQuantop and denotes the doubly-variable-basis category described as follows: 1. Objects. (G, M, R, L), where (G, M, R) is an L-formal context. 2. Morphisms. ( f, ) : (G 1 , M1 , R1 , L 1 ) → (G 2 , M2 , R2 , L 2 ), where f : L 1G 1 → L 2G 2 , op : L 2M2 → L 1M1 are mappings satisfying H1 = op ◦ H2 ◦ f,

K 2 = f ◦ K 1 ◦ op ,

where (H1 , L 1G 1 , L 1M1 , K 1 ) and (H2 , L 2G 2 , L 2M2 , K 2 ) are the Galois connections respectively generated by (G 1 , M1 , R1 ) and (G 2 , M2 , R2 ). In future work, application oriented motivations and functorial relationships of AbRSQuantop -FCI to (doubly-) variable-basis Chu systems, Loc-TopSys, etc., will be developed. Finally, the authors commend one of the referees for a meticulous review of the original submission which enabled the authors to make many improvements resulting in the present work.

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

89

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

[31]

[32] [33] [34] [35] [36] [37] [38] [39]

[40]

J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, 2nd ed., Dover Publications, New York, 2009. M. Barr, ∗ -Autonomous Categories, in: Lecture Notes in Mathematics, vol. 752, Springer Verlag, Berlin/Heidelberg/New York, 1979. R. Bˇelohlávek, Fuzzy Galois connections, Math. Logic Q. 45 (4) (1999) 497–504. R. Bˇelohlávek, Fuzzy Galois connections and fuzzy concept lattices: from binary relations to conceptual structures, Discovering the World with Fuzzy Logic, Studies in Fuzziness and Soft Computing, vol. 57, Physica, Heidelberg, 2000, pp. 462–494. R. Bˇelohlávek, Lattices of fixed points of fuzzy Galois connections, Math. Logic Q. 47 (2001) 111–116. R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, Boston, Dordrecht, London, New York, 2002. R. Bˇelohlávek, What is a fuzzy concept lattice? in: Proceedings of the CLA 2005, 7–9 September 2005, Olomouc, Czech Republic, pp. 34–45. G. Birkhoff, Lattice Theory, 1st ed., vol. 25, American Mathematical Society Colloquium Publications, 1940. G. Birkhoff, Lattice Theory, 3rd ed., vol. 25, American Mathematical Society Colloquium Publications, 1967. A. Burusco, R. Fuentes-Gonzalez, Concept lattices defined from implication operators, Fuzzy Sets Syst. 114 (3) (2000) 431–436. J.T. Denniston, A. Melton, S.E. Rodabaugh, Lattice-valued predicate transformers and interchange systems, in: P. Cintula, E.P. Klement, L.N. Stout (Eds.), Abstracts of the 31st Linz Seminar, Universitätsdirecktion Johannes Kepler Universität, Linz, Austria, February 2010, pp. 31–40. J.T. Denniston, A. Melton, S.E. Rodabaugh, Interweaving algebra and topology: lattice-valued topological systems, Fuzzy Sets Syst. 192 (2012) 58–103. J.T. Denniston, A. Melton, S.E. Rodabaugh, Lattice-valued predicate transformers and lattice-valued Chu systems, in submission. J.T. Denniston, S.E. Rodabaugh, Functorial relationships between lattice-valued topology and topological systems, Quaest. Math. 32 (2) (2009) 139–186. E.W. Dijkstra, A Discipline of Programming, Prentice-Hall, Englewood Cliffs, NJ, 1976. M. El-Zekeya, V. Novak, R. Mesiar, On good EQ-algebras, Fuzzy Sets Syst. 178 (2011) 1–23. M. Erné, J. Koslowski, A. Melton, G.E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Ann. N.Y. Acad. Sci., vol. 704, 1993, pp. 103–125. A. Frascella, Fuzzy Galois connections under weak conditions, Fuzzy Sets Syst. 172 (2011) 33–50. N. Galatos, Equational bases for joins of residuated-lattice varieties, Studia Logica 76 (2004) 227–240. N. Galatos, Generalized ordinal sums and translations, Logic J. IGPL 19 (2010) 455–466. N. Galatos, P. Jipsen, Residuated frames with applications to decidability, Trans. Amer. Math. Soc., in press. N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: an algebraic glimpse at substructural logics, Studies in Logics and the Foundations of Mathematics, vol. 151, North-Holland (Elsevier), Amsterdam, 2007. N. Galatos, H. Ono, Cut elimination and strong separation for substructural logics: an algebraic approach, Ann. Pure Appl. Logic 161 (9) (2010) 1097–1133. B. Ganter, G. Stumme, R. Wille, Formal concept analysis: foundations and applications, in: Lecture Notes in Artificial Intelligence, vol. 3626, Springer Verlag, Berlin, Heidelberg, New York, 2005. B. Ganter, R. Wille, Formale Begriffsanalyse: Mathematishce Grundlagen, Springer Verlag, Berlin, Heidelberg, New York, 1999 (German edition 1996/English edition). J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. C. Guido, Fuzzy points and attachment, Fuzzy Sets Syst. 161 (10) (2010) 2150–2165. C. Guido, Characterization of functions among lattice-valued relations, preprint, 2010. Sheng-Wei Han, Bin Zhao, Nuclei and conuclei on residuated lattices, Fuzzy Sets Syst. 172 (2011) 51–70. U. Höhle, Commutative, residuated l-monoids, in: U. Höhle, E.P. Klement (Eds.), Non-Classical Logics and Their Application to Fuzzy Subsets, Series B: Mathematical and Statistical Methods, vol. 32, Kluwer Academic Publishers, Boston, Dordrecht, London, New York, 1995, pp. 53–106. U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, New York, 1999, pp. 123–272, (Chapter 3). P. Jipsen, C. Tsinakis, A survey of residuated lattices, in: J. Martinez (Ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, Boston, Dordrecht, London, New York, 2002, pp. 19–56. P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. Y. Lafont, T. Streicher, Games semantics for linear logic, in: Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, July 1991, pp. 43–49. A. Melton, D.A. Schmidt, G.E. Strecker, Galois connections and computer science applications, in: Lecture Notes in Computer Science, vol. 240, Springer Verlag, Berlin, Heidelberg, New York, 1986, pp. 299–312. J.M. McDill, A.C. Melton, G.E. Strecker, A category of Galois connections, in: Lecture Notes in Computer Science, vol. 283, Springer Verlag, Berlin, Heidelberg, New York, 1987, pp. 290–300. O. Ore, Galois connexions, Trans. Am. Math. Soc. 55 (1944) 493–513. V.R. Pratt, Chu spaces, http://chu.standford.edu/, 30 September 2011. S.E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology: preliminary report, in: E.P. Klement (Ed.), Proceedings of the Third International Seminar on Fuzzy Set Theory, vol. 3, Universitätsdirektion Johannes Kepler Universtät, A-4040, Linz, Austria, 1981, pp. 119–152. S.E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy Sets Syst. 9 (1983) 241–265.

90

J.T. Denniston et al. / Fuzzy Sets and Systems 216 (2013) 52 – 90

[41] S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, New York, 1999, pp. 273–388, (Chapter 4). [42] S.E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics, Int. J. Math. Math. Sci. (3) (2007), Article ID 43645, 71 pp., http://dx.doi.org/10.1155/2007/43645, http://www.hindwai.com/gearticle.aspx?. [43] S.E. Rodabaugh, Necessity of non-stratified and anti-stratified spaces in lattice-valued topology, Fuzzy Sets Syst. 161 (2010) 1253–1269. [44] K.I. Rosenthal, Quantales and their applications, Pittman Research Notes in Mathematics, vol. 234, Addison Wesley Longman, Burnt Mill, Harlow, 1990. [45] J. Schmidt, Beiträge zur filtertheorie II, Math. Nachr. 10 (1953) 197–232. [46] S.A. Solovyov, Variable-basis topological systems versus variable-basis topological spaces, Soft Computing 14 (10) (2010) 1059–1068. [47] S.A. Solovyov, Categorical foundations of variety-based topology and topological systems, Fuzzy Sets Syst., in press, http://dx.doi.org/10.1016/j.fss.2011.07.016. [48] S.A. Solovyov, On lattice-valued Formal Concept Analysis, in submission. [49] G. Stumme, Efficient data mining based on formal concept analysis, in: A. Hameurlain, R. Cicchetti, R. Traunmüller (Eds.), Database and Expert Systems Applications, Lecture Notes in Computer Science, vol. 2453, Springer Verlag, Berlin, Heidelberg, New York, 2002, pp. 534–546. [50] Esko Turunen, Algebraic structures in fuzzy logic, Fuzzy Sets Syst. 52 (1992) 181–188. [51] R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (Ed.), Ordered Sets, Reidel, Boston, Dordrecht, 1982, pp. 445–470. [52] R. Wille, Formal concept analysis as mathematical theory of concepts and concept hierarchies, in: B. Ganter, G. Stumme, R. Wille (Eds.), Lecture Notes in Artificial Intelligence, vol. 3626, Springer Verlag, Berlin, Heidelberg, New York, 2005, pp. 1–33 (Chapter 1). [53] S.J. Vickers, Topology Via Logic, Cambridge University Press, Cambridge, 1989. [54] B. Vormbrock, R. Wille, Semiconcept and protoconcept algebras: the basic theorems, in: B. Ganter, G. Stumme, R. Wille (Eds.), Lecture Notes in Artificial Intelligence, vol. 3626, Springer Verlag, Berlin, Heidelberg, New York, 2005, pp. 34–48 (Chapter 2).