Fuzzy points and attachment

Fuzzy Sets and Systems 161 (2010) 2150 – 2165 www.elsevier.com/locate/fss

Fuzzy points and attachment Cosimo Guido∗ Department of Mathematics, “E. De Giorgi”-University of Salento, P.O. Box 193-73100 Lecce, Italy Received 19 January 2009; received in revised form 24 February 2010; accepted 26 February 2010 Available online 7 March 2010

Abstract In this paper a binary relation between L-sets, called attachment, is defined by means of a suitable family of completely coprime filters in L. Examples are given using several kinds of lattice-ordered algebras and a class of attachments is determined whose elements generalize Pu–Liu’s quasi-coincidence relation. Mapping any L-set on X to the set of the attached L-points one gets a frame map from the L-powerset L X to the powerset P(S X ) of the set S X of L-points on X. This allows one to define functors from the category L-Top of L-topological spaces either to the category Top of topological spaces or to the category TopSys of topological systems. A comparison with the hypergraph functor is also included. © 2010 Elsevier B.V. All rights reserved. Keywords: Lattice-valued mathematics; L-topological space; Fuzzy point; Attachment; Quasi-coincidence; Topological system; Hypergraph functor

1. Introduction Topological concepts involving fuzzy points and fuzzy subsets have been among the notions considered in the field of fuzzy set theory almost from its inception (see [1] and [32]). This is not surprising since order-theoretic and latticetheoretic basic notions are the fundamental tools to deal with in fuzzy set theory and their immediate applications bring to approach topological spaces and in particular points, neighborhoods, convergence, compactness and separation in the fuzzy context. The ensuing issues of motivation and justification of topological structure have been considered from various points of view, starting from the time when only L-topological spaces (specifically [0,1]-topological spaces) were under study. There were at least three fundamental approaches to treating L-topological spaces as non-trivial extensions of ordinary topological spaces. The first one is functorial and it started with the work of Lowen [21,22] whose functors allowed the evaluation of “goodness” of topological concepts. Subsequently and more generally, a category-theoretic approach to lattice-valued topology has been developed and systematically organized, mainly by Rodabaugh [25–27]. The second approach involves points and local aspects of topological spaces, including convergence, and may be linked to the fundamental question of characterizing topology by neighborhood systems [3,23,32] or more generally to the study of relations between fuzzy points (fuzzy sets whose support is a singleton) and open fuzzy sets. The third approach treats the matter without recourse to points; here the structure of the lattice within which the sublattice of open fuzzy sets is embedded has a crucial role. This, of course, is the case of completely distributive lattices with an order reversing involution, now called Hutton algebras (see [16,17,31]). ∗ Tel.: +39 0832 297428; fax: +39 0832 297410.

E-mail address: [email protected]. 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.02.009

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As is evident from the considerable body of literature using the lattice-theoretic point of view, it is important to analyze links between the topological concepts in classical mathematics and the new tools and techniques of manyvalued reasoning, including (mainly) algebraic aspects of non-classical logics and their applications. This paper has the primary goal of providing tools that permit description of L-topological spaces as spaces of points. This motivates and formalizes, quite generally, the approach to fuzzy topological spaces organized and presented by Pu and Liu [23]. This approach, which depends crucially on the quasi-coincidence relation, has been used widely in both theoretical investigations and applications, especially in the area of fuzzy dynamical systems. The quasi-coincidence relation depends on the rather strong structure of an MV-chain on the unit interval, as we shall see. In this paper a more general relation called attachment is introduced, examples are given, and fundamental properties which depend on the structure of the range lattice L are elucidated. In fact several kinds of algebras frequently used in many-valued mathematics and logics are considered. Among other results it emerges that if L is a spatial locale, then any L-topological space (X, ) can be viewed as a topological space whose points are the L-points on X and whose topology is frame-isomorphic to : so this latter space completely describes the local structure of (X, ). Such a result seems to justify the opinion sometimes expressed (see [12,15]) that L-topological spaces merit consideration only in case L is not a spatial locale: otherwise there would be nothing new with respect to the classical context. Be this as it may, and even when L is spatial, the attachment view yields insight into • how L-topological spaces can be viewed as topological spaces; • how topological concepts can be interpreted in an L-valued context; • which L-topological properties are relevant, and how those can be formulated. In this respect it can be seen that dealing with an L-topological space (even for a spatial frame L) gives something new and non-trivial. Examples of the realization of this possibility are given in [9]. As is shown in Section 4 below, the attachment relation permits the construction of functors that illuminate the relationship between lattice-valued topology and topological systems (see [5]). Many-valued extension of the attachment relation and links to many-valued filters and topologies are outlined in Section 5, where the strict link between attachment and the hypergraph functor is also analyzed. 2. Preliminaries In this paper we follow the standard notation of lattice-valued mathematics (see, e.g. [7,12,14,18,20,28]); we recall here only some definitions and concepts and state a few variations to the usual notation. (L ≤) or simply L will always denote a complete lattice, whose algebraic structure is completely determined, as is well known, by the order relation on the underlying set; ⊥ and  denote the bottom and the top element respectively. The terms (semi-)frame, (semi-)locale and the corresponding categories (S)Frm, (S)Loc are used as in [18,26,27] whose notation and terminology we also follow for the notions of prime, coprime, completely coprime elements, ideals, filters, points in a locale, spatiality and so on. We change the terminology only for the complements of principal prime ideals in a complete lattice, which we define as follows: Definition 2.1. A completely coprime filter in L is a filter F such that for every S ⊆ L the implication  S ∈ F ⇒ S ∩ F ∅ is true. In fact, in [7,18] such filters are said to be completely prime. We prefer “completely coprime” because these filters are of the kind ↑ b, with b  ⊥ completely coprime, whenever they are principal; they have nothing to do with completely prime elements. Note that a completely coprime filter is necessarily proper. A coframe  is a complete lattice whose opposite is a frame, so it is characterized by the distributivity condition a ∨ ( S) = {a ∨ s|s ∈ S} for all a ∈ L, S ⊆ L; of course if (L , ≤) has an order reversing involution then it is a frame if and only if it is a coframe.

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We shall not use the term “algebra” to denote those structures which are simply lattices, i.e. objects whose operations are completely determined by the order relation: so we shall speak of (complete) boolean lattices and say that the objects of Frm and Loc are complete Heyting lattices. We use the term “algebra” exclusively for those lattice-ordered structures that are not determined by the order relation only, e.g. MV-algebras, BL-algebras, MTL-algebras, quantales and others, including (complete) residuated algebras, usually called (complete) residuated lattices. These algebraic structures, mostly arising from non-classical, mainly many-valued, logics, are frequently used in lattice-valued mathematics (see [2,11]). As is shown in [10] the structure of these algebras depends completely on an underlying extended-order, which is a binary operation in L, and a value true, which is a 0-ary operation in L. As we shall need these structures in some important examples, we recall a few definitions and results from [10]. Definition 2.2. A complete, distributive extended-order algebra (cdeo algebra) is a (2,0)-type algebra (L , →, ) such that • the binary relation ≤ in L defined by x ≤ y if and only if x → y =  is a partial order; • (L a complete  , ≤) is  lattice with largest element  (⊥ denotes the least element); • A → B = ( A → B), for all A, B ⊆ L where A → B = {a → b | a ∈ A, b ∈ B}. We note that in a cdeo algebra (L , →, ) the following hold for all a, b, c ∈ L • if a → b =  then (c → a) → (c → b) =  (weak isotonic condition); • if a → b =  then (b → c) → (a → c) =  (weak antitonic condition). Moreover the adjoint product ⊗ : L × L → L can be defined by  a⊗b = {x ∈ L|b ≤ a → x} In this paper (L , →, ) will denote a cdeo algebra with adjoint product ⊗; further conditions on L, when needed, will be explicitly stated. It is also worth noting, as is shown in [10], that the completeness assumption in this class of algebras is not restrictive. Proposition 2.3. In any cdeo algebra (L , →, ) the following hold for all a, b, c ∈ L and for all A, B ⊆ L 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

a ⊗ b ≤ a, a ⊗  = a, a ⊗ ⊥ = ⊥, ⊥ ⊗ a = ⊥,  ⊗ a ≤ a if and only if  → a ≥ a,  ⊗ a≥ a if and  only if (∀x ∈ L :  → x ≥ a ⇒ x ≥ a), a ⊗ ( B) = (a ⊗ B), Ifb ≤ c, then  a ⊗ b ≤ a ⊗ c, ( A) ⊗ b ≥ (A ⊗ b), If a ≤ b, then a ⊗ c ≤ b ⊗ c, a ⊗ (a → b) ≤ b ≤ a → (a ⊗ b).

Definition 2.4. A cdeo algebra (L , →, ) is commutative if (c) a → (b → c) =  ⇔ b → (a → c) =  (weak exchange condition). L is associative if either of the following, equivalent conditions is satisfied  (a1 ) a → (b → c) = ( {x|a → (b → x) = }) → c, (a2 ) a → (b → c) = (b ⊗ a) → c.

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Proposition 2.5. (L , →, ) is commutative (associative) if and only if the adjoint product ⊗ is commutative (associative). Proposition 2.6. Let (L , →, ) be a cdeo algebra and let ⊗ be its adjoint product. If (L , →, ) is commutative, then: 1. 2. 3. 4. 5. 6. 7.

 is the neutral element for ⊗; a ⊗ b ≤ b ≤ a → b; a ⊗ b ≤ a ∧ b; a ≤ b → (a ⊗ b); a ≤ (a → b) → b; → a = a;  ( A) ⊗ b = (A ⊗ b).

If (L , →, ) is associative, then: 8. (a → b) ⊗ (b → c) ≤ a → c; 8 . (b → c) → ((a → b) → (a → c)) =  (isotonic condition). If (L , →, ) is associative and commutative, then: 9. a → (b → c) = b → (a → c) (exchange condition); 10. (a → b) → ((b → c) → (a → c)) =  (antitonic condition); 11. a ⊗ (b → c) ≤ b → (a ⊗ c). Clearly, a commutative and associative cdeo algebra is nothing but a complete residuated algebra or, using the terminology of [15], a strictly two sided commutative quantale. (L , →, ) is a Heyting algebra if the adjoint product is just ⊗ = ∧. It is important to note that (L , ≤) may be a complete Heyting lattice (i.e. a frame) also in cases when (L , →, ) is not a Heyting algebra; in fact complete MV-algebras (complete residuated algebras satisfying the conditions (1) a ⊗ (a → b) = a ∧ b, (2) (a → b) ∨ (b → a) = , (3) (a → ⊥) → ⊥ = a) are complete Heyting lattices (i.e. frames) but they are not Heyting algebras. With respect to fuzzy set theory and fuzzy topology we adopt the framework for the fixed-basis context described in [15,26,27]. Since we use a quite general complete lattice L we usually prefer to speak of L-sets, L-topological spaces and so on, instead of using the prefix “fuzzy”; then an L-topology  on X ( closed under finite infs and arbitrary sups in L X ) need not be a frame. For a fixed lattice L and a fixed set X we denote by Y ,  ∈ L, Y ⊆ X , the L-set on X taking value  on the elements y ∈ Y and value ⊥ elsewhere. x stands for {x} and denotes the L-point on X with support x and value ; S X denotes the set of L-points on X. We adopt the arrow notation for the L-powerset operators and for the classical powerset operators, as well, as in [5,8,26,27].

3. The attachment relation Definition 3.1. An attachment family, or more simply an attachment in a complete lattice L is a family A = {F | ∈ L} of subsets of L with • F⊥ = ∅; • F a completely coprime filter ∀  ⊥. Definition 3.2. The attachment is said to be • spatial if (sp)  ⱕ  ⇒ ∃ :  ∈ F ,  ∈ / F ; • isotonic if (i)  ≤  ⇒ F ⊆ F ;

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• symmetrical if (s)  ∈ F ⇒  ∈ F ; • inverse isotonic if (ii) F ⊆ F ⇒  ≤ . The following relationships among the conditions listed in the above definition holds. Proposition 3.3. For any attachment A in L one has • (s) ⇒ (i); • (s), (ii) ⇒ (sp). Proof. Assuming (s), the following implications are true and prove (s) ⇒ (i)  ≤ ,  ∈ F ⇒  ≤ ,  ∈ F ⇒  ∈ F ⇒  ∈ F . Now, assume (s) and (ii) and consider ,  ∈ L ,  ⱕ ; then by (ii) F F hence  exists in L such that / F and by symmetry  ∈ F ,  ∈ / F .   ∈ F ,  ∈ Remark 3.4. Let L ⊥ = L \ {⊥}. Then • If A is a spatial attachment in L the equalities   F = L ⊥ and F = {} ∈L

∈L ⊥

/ F . are true. In fact, by spatiality, if  ⱕ ⊥ and  ⱕ  then ,  ∈ L exist such that  ∈ F ,  ∈ • If A is isotonic and spatial then F = L ⊥ . In fact F = ∈L F = L ⊥ . • If A satisfies (s) and (sp) the equivalence F = L ⊥ ⇔  =  holds. In fact since (s) ⇒ (i) it is also true that F = L ⊥ . Conversely, if  ⱕ , then by spatiality  ∈ L ⊥ exists s.t. / F and F  L ⊥ . ∈ / F , hence, by symmetry,  ∈ We shall list several examples of attachments related to some lattice-ordered algebras used in lattice-valued mathematics; some of them show that the implications in Proposition 3.3 above are not reversible. We note also that a complete lattice that has an attachment has primes: in fact the complement of a completely coprime filter is a principal prime ideal. Conversely, any complete lattice with primes has an attachment as is shown below. Example 3.5. If L has a prime element a ∈ L, there is an attachment given by setting F⊥ = ∅ and F = {|  ⱕ a} if   ⊥; evidently A = {F | ∈ L} may not satisfy either (s) or (ii) or (sp), in general. Note that, as is also shown in [5], adding to any complete lattice a new bottom element gives a new complete lattice whose (new) bottom is prime. For any attachment A, a binary relation also denoted by A, called the attachment relation, is defined between L-sets on any set X by A A B if B(x) ∈ FA(x) for some x ∈ X Usually we shall simply speak of attachment when referring either to the family A or to the relation A. The attachment relation can be specialized for L-points as follows: Remark 3.6. Let x ,  y ∈ S X and A ∈ L X , • x AA ⇐⇒ A(x) ∈ F ; • AAx ⇐⇒  ∈ FA(x) ; • x A y ⇐⇒ x = y and  ∈ F . Then the following can be easily proved.

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Proposition 3.7. For all A, B ∈ L X , the equivalences below are true • A A B ⇔ ∃x ∈ S X such that A A x and x ≤ B. if the attachment is isotonic • AAB ⇔ ∃x ≤ A, x ≤ B such that x A x . Moreover the following easy results will be useful. Lemma 3.8. With obvious notation, for any attachment A and any non-empty set X one has • • • • • •

A A B, B ≤ B  ⇒ A A B  ; x A  X , ∀x ∈ S X ;   x AA,  x AA ⇔ x A A ∧ A ; x A i∈I Ai ⇔ ∃i ∈ I : x A Ai ;   AA x , AAx ⇔ AA( ∧  )x ; AA( i∈I i )x ⇔ ∃i ∈ I : AA(i )x

if the attachment is isotonic, then • AAB, A ≤ A ⇒ A AB. If the relation A is restricted to the product S X × L X then we consider the right cones A( y ),  y ∈ S X , and the left cones, A− (A), A ∈ L X , of A; to simplify notation we shall denote the left cone by A∗ , A ∈ L X , so we have A( y ) = {B ∈ L X | y AB} and A∗ = {x ∈ S X |x A A} Then the following properties hold: Proposition 3.9. For any attachment A in L and for all the considered L-points and L-sets on X one has 1. 2. 3. 4.

A( y ) is a completely coprime filter in L X ; A ≤ B ⇒ A∗⊆ B ∗ ; if J is finite: ( j∈J A j )∗ = j∈J A∗j ;   ( i∈I Ai )∗ = i∈I Ai∗ .

Proof. 1. B ∈ L X is in A( y ) if and only if B(y) is in F , which is a completely coprime filter; the assertion then follows directly from Lemma 3.8. 2, 3. These follow since F is a filter for every L-point x . 4. This holds since F is completely coprime.  Proposition 3.10. For any spatial attachment A in L and for all A, B ∈ L X , x ∈ X the following equivalences are true 1. 2. 3. 4.

A(x)  ⊥ ⇐⇒ ∃x ∈ S X : x A A; A(x) =  ⇐⇒ x A A, ∀x ∈ S X ; A∗ ⊆ B ∗ ⇐⇒ A ≤ B; A∗ ∩ B ∗ = ∅ ⇐⇒ A ∧ B = ⊥ X .

Proof. 1, 2. follow easily by Remark 3.4. 3. By Proposition 3.9 we only need to prove that A ≤ B under the assumption A∗ ⊆ B ∗ . In case A ⱕ B and, in particular, A(x) ⱕ B(x) the assumed condition (sp) gives  ∈ L such that A(x) ∈ F , B(x) ∈ / F , hence x ∈ A∗ \ B ∗ ∗ ∗ and consequently A B . 4. A(x) ∧ B(x) ⱕ ⊥ ⇒ ∃ : A(x) ∧ B(x) ∈ F ⇒ A(x) ∈ F , B(x) ∈ F ⇒ x ∈ A∗ ∩ B ∗ . Conversely, x ∈ A∗ ∩ B ∗ ⇒ A(x) ∈ F , B(x) ∈ F ⇒ A(x) ∧ B(x) ∈ F ⇒ A(x) ∧ B(x)  ⊥. 

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Now fix an attachment and consider the function ∗ : L X −→ P(S X ), AA∗ Proposition 3.11. For any fixed attachment 1. * is a frame map; under (sp) 2. * is an embedding of posets. Proof. 1. follows immediately from 3 and 4 of Proposition 3.9. Now, assuming (sp), since * preserves infs (and sups, as well) and is injective by item 3 of Proposition 3.10, it is easy to see that * preserves and reflects the order relation; and statement 2 follows.  The following is a quite general, well known result in lattice theory and it can easily be proved. Lemma (M, ≤) be complete lattices, f : L → M a frame map and let S ⊆ L be closed under 3.12. Let (L , ≤) and  ∧ and in L (i.e. (S, ≤) is a -complete sublattice of (L , ≤)). Then the following hold   • If S ⊆ L is a frame, (S, ≤), with ∧ and induced by L, then ( f → (S), ≤) is a frame with ∧ and induced by M. • If (M, ≤) is a frame then ( f → (S), ≤) is a subframe of M. Let us denote, for any set X ∗X = ∗→ (L X ) = {A∗ | A ∈ L X } Then the following holds. Proposition 3.13. For any attachment A in L 1. ∗X is a topology on S X ; if the attachment is spatial then 2. (∗X , ⊆) and (L X , ≤) are isomorphic frames, for any non-empty set X; 3. L is a frame. Proof. 1. Follows directly from Lemma 3.12. 2. By Proposition 3.11, ∗ : L X → ∗ is a surjective embedding of posets, hence it is a frame isomorphism. 3. Follows from item 2 since the L-powerset L {x} of a singleton {x} is isomorphic to L.  We now show that a spatial attachment exists in L if and only if L has enough primes. We denote by P(L) the subset of prime elements of L. Proposition 3.14. For any complete lattice L the following are equivalent 1. A spatial attachment exists in L. 2. L is a spatial frame. 3. ∀ ⱕ  a prime  ∈ L exists s.t.  ≤ ,  ⱕ . Proof. 1. ⇒ 2. If A is a spatial attachment, then by Proposition 3.13 L is a frame. Now, let ,  ∈  L,  ⱕ  and let / F for some  ∈ L. Then the point p ∈ pt(L) corresponding to the prime element (L \ F  ∈ F ,  ∈  ) clearly satisfies the condition  p() = ⊥. Moreover p() = : otherwise, it would follow from p() = ⊥ that  ≤ (L \ F ) / F , a contradiction. and since L \ F =↓ ( (L \ F )) it should follow that  ∈ 2. ⇒ 3. If  ⱕ  then, by the assumption, p ∈ pt(L) exists such that p() =  and p() = ⊥. Since the kernel of p is a principal prime ideal, p← (⊥) =↓  for some  ∈ P(L) and consequently  ≤  and  ⱕ .

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3. ⇒ 1. Put F⊥ = ∅, F = { ∈ L| ⱕ } if  ∈ P(L) \ {⊥}. Moreover, fix any ¯ ∈ P(L) and put F = F¯ if  ∈ (L \ P(L)) \ {⊥}. Then {F | ∈ L} is a spatial attachment. In fact F is a completely coprime filter, ∀  ⊥; moreover, if  ⱕ ,  ∈ P(L) exists such that  ≤  and  ⱕ , hence  ∈ / F and  ∈ F .  A spatial attachment need not be either isotonic, or symmetrical or inverse isotonic, as the following example shows. Example 3.15. Let L be the cofinite topology on an infinite set S. L, of course, is a spatial frame whose primes are the complements of the singletons and the empty set. Fix any x0 ∈ S and construct F ,  ∈ L, as follows: if  = S \ {x } then F = { ∈ L| x ∈ }; if |S \ | ≥ 2 then F = { ∈ L|x0 ∈ } F∅ = ∅; FS = L \ {∅}. As is shown in the proof of Proposition 3.14, A = {F | ∈ L} is a spatial attachment. However, for x  x 0 in S,  = S \ {x, x0 },  = S \ {x} and  = S \ {x0 } one has  ⊆  but F F ; moreover F = F but  . So A does not satisfy either (i) or (ii) (or (s), of course, by Proposition 3.3). Example 3.16. Let L be a complete lattice with an order reversing involution  : L → L and suppose that all the elements of L, except  are prime (the results and discussion above clearly show that L is a frame and a coframe). It is easy to verify that the family A = {F | ∈ L}, with F = {| ⱕ ()} is an attachment that satisfies (i) and (s). It then follows from  ⱕ  that () ⱕ (), so () belongs to F but evidently it does not belong to F , hence F F and (ii) is satisfied; (sp) follows from (ii) and (s) by Proposition 3.3. Example 3.17. Let L be a complete lattice. The family A = {F | ∈ L}, with F⊥ = ∅ and F = {}, ∀  ⊥, is an attachment family if and only if  is completely coprime. If this is the case, then A satisfies (i) and, moreover, it satisfies (s) or (ii) or (sp) if and only if L = {⊥, }. Interesting examples of attachment arise from lattice-ordered multiplicative structures. As already noted in Section 2, most of those can be viewed as extended-order algebras. Example 3.18. Let (L , →, ) be a cdeo algebra whose product ⊗ is distributive over finite infs on the right and let a be a prime element of the underlying lattice L. The family A = {F | ∈ L}, with F = {| ⊗  ⱕ a} is an isotonic attachment. Since the product may be not commutative, (s) is not satisfied, in general. (ii) and (sp) may not be satisfied either as the examples below demonstrate. Example 3.19. Let (L , →, ) be a cdeo algebra, let ¬ =  → ⊥ define the negation and assume that ¬ is prime, ∀  ⊥. Let F = {| ⱕ ¬} = {| ⊗  > ⊥}. The family A = {F | ∈ L} is an isotonic attachment and it satisfies the condition (s) if and only if  ≤ ¬¬, ∀ ∈ L. In fact, assuming the latter condition, if ,  ∈ L then the following implications are true  ∈ / F ⇒  ≤ ¬ ⇒ ¬ ≥ ¬¬ ≥  ⇒  ∈ / F . Conversely, assuming (s) means that the equivalence ⊗ = ⊥ ⇐⇒ ⊗ = ⊥ is true. Then, in particular, it follows from  ⊗ ¬ = ⊥ that ¬ ⊗  = ⊥ hence  ≤ ¬ → ⊥ = ¬¬. Note that the condition  ≤ ¬¬, which is equivalent to ¬ ⊗  = ⊥, is satisfied if the product is commutative: in fact the equality  ⊗ ¬ = ⊥ is true in L in any case. The attachment just defined may be neither spatial nor inverse isotonic as the next example shows. Example 3.20. Let P be the product algebra ([0, 1], ∨, ∧, ⊗ P , → P , 0, 1), with a ⊗ P b = a · b and ⎧ ⎨ 1 if a  b a →P b = b ⎩ otherwise a ([0, 1], → P , 1) is a commutative, associative cdeo algebra, of course. We consider ∀ ∈ [0, 1], F = {| ⊗ P  > 0} = {| ·  > 0}. A = {F | ∈ [0, 1]} is an isotonic, symmetrical attachment. However A does not satisfy either (ii) or (sp): in fact F = (0, 1] for every   0.

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Note that ¬a = 0, ∀a  0, in this algebra. Also note that this attachment is not spatial though the lattice is a spatial frame (not a Heyting algebra, however) and it is not inverse isotonic, of course. Example 3.21. Let L be a frame. The family A = {F | ∈ L}, with F = {| ∧  > ⊥} is an isotonic, symmetrical attachment provided that ⊥ is prime. In this case, since F = L \ {⊥} for every   ⊥, A does not satisfy either (ii) or (sp) unless L = {⊥, }. Note that in this special kind of cdeo algebra (⊗ = ∧) assuming that ¬ is prime, ∀  ⊥, as in Example 3.19 can be relaxed to assuming that ⊥ is prime with the same results. Example 3.22. Let L be the Łukasiewicz algebra ([0, 1], ∨, ∧, ⊗ L , → L , 0, 1), with a ⊗ L b = max{a + b − 1, 0} and a → L b = min{1 − a + b, 1}. The family A = {F | ∈ L}, where F = {| ⊗ L  > 0} = {| > 1 − } is an attachment that satisfies (ii) and (s), and therefore also (i) and (sp). We note that this example falls within the same framework as Examples 3.16, 3.18 and 3.19. Also note that this attachment depends on the algebraic structure of L, which is not a Heyting algebra, though the underlying lattice is a (spatial) frame. The attachment shown in Example 3.22 is exactly the quasi-coincidence relation introduced in [23] and quite frequently used in the theory and application of fuzzy topology. All the results presented in this paper and in [9] apply, of course, to the quasi-coincidence based approach to [0,1]-topological spaces presented in [23] and give motivation to that approach and to most results consequently obtained; nevertheless, some notions and some results based on quasi-coincidence are shown in [9] to be not well motivated. Among the examples just given, (3.16) and (3.22) are the only ones, apart from the trivial case when L = {⊥, }, that provide attachments satisfying (ii) and (s), hence (i) and (sp). Proposition 3.24 below explains why. Lemma 3.23. If A satisfies (s) then for all i ∈ L , i ∈ I , the equality  Fi Fi∈I i = i∈I

holds. Proof. x ∈ Fi∈I i ⇔

 i∈I

i ∈ Fx ⇔ ∃ j ∈ I :  j ∈ Fx ⇔ ∃ j ∈ I : x ∈ F j ⇔ x ∈

 i∈I

Fi . 

Now we can prove the following result. Proposition 3.24. Let L be a complete lattice. An attachment A exists in L that satisfies (ii) and (s) if and only if an order reversing involution exists in L and all the elements of L other than  are prime. Proof. That the stated conditions ensure the existence of anattachment with (ii) and (s) has been shown in Example 3.16. To show the converse define, for every a ∈ L, (a) = (L \ Fa ). Then  • (⊥) = L = ; • it follows from a  ⊥ that  ∈ Fa hence, by symmetry, a ∈ F ; then F = L \ {⊥} and hence () = ⊥; • if ⊥  a  , then  a= (L \ F(L\Fa ) ) = ((a))  Fa ) ∈ / Fb ⇒ (L \ F Fb= ∅ ⇒ Fb ⊆ in fact the following implications are true b ∈ L \ F(L\Fa ) ⇒ (L \  a)   whenever b ∈ L \ F  ≥ ⇒ b ≤ a; moreover, if b ≤ a then a (L \ F ) = (L \ F a (L\F (L\F ) ) x∈L\Fa Fx ) = a a   ( x∈L\Fa (L \ Fx )) = ( a∈L\Fx (L \ Fx )) ≥ a;   • by Proposition 3.3 A satisfies (i), hence it is clear that if a ≤ b then (b) = (L \ Fb ) ≤ (L \ Fa ) = (a); • finally, it is well known that the complement of a completely coprime filter is a principal prime ideal, hence L \ Fa =↓ (a) and (a) is prime, ∀a ∈ L ⊥ ; so every b ∈ L other than  is prime since  is an involution. 

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We note that attachments satisfying (ii) and (s) can be obtained only as in Example 3.16 by means of the order reversing involution determined as in the proof of Proposition 3.24. In fact with the above notation, since L \ F =↓ () then F = L\ ↓ () = {| ⱕ ()}. One can say that (ii) and (s) characterize attachments that can be considered as generalizations of the quasicoincidence relation, which suggests: Definition 3.25. A quasi-coincidence attachment (shortly q-attachment) is an attachment that satisfies (ii) and (s). It is well known that for any lattice L the equivalence every a ∈ L  is prime ⇐⇒ L is a chain holds. Hence, after recalling that a deMorgan algebra is a complete lattice with an order reversing involution and a Hutton algebra is a completely distributive lattice with an order reversing involution (see [27]), the following is an immediate consequence of Proposition 3.24, taking into account that every complete chain is completely distributive. Corollary 3.26. For any complete lattice L the following are equivalent • L has a q-attachment; • L is a deMorgan chain; • L is a Hutton chain. 4. Functors defined by attachment For any L-topological space (X, ) and for any fixed attachment denote ∗ = {A∗ |A ∈ }. Then clearly (S X , ∗ ) is a topological space. Moreover,  and ∗ are isomorphic frames if the attachment is spatial. These statements extend Proposition 3.13 and can be proved similarly by setting  X to L X (the discrete L-topology on X) to conform with the notation used in the proposition. Denote by L-T (X ) the set of all L-topologies on X and by T ∗ (S X ) = {∗ |  ∈ L-T (X )} the set of all corresponding topologies on S X , both ordered by setinclusion. Then it is known that L-T (X ) is a -subsemilattice of P(L X ), hence a complete lattice.  * only preserves , in general, the Since the forward powerset operator ∗→ : P(L X ) −→P(P(S X )) of the function  image T ∗ (S X ) = (∗→ )→ (L-T (X )) need not be either a -subsemilattice or a -subsemilattice of P(P(S X )); in fact, it may not be a lattice at all.   However, if A is a spatial attachment then * is injective and consequently ∗→ is injective and it preserves and . Hence in this case ∗→ : L-T (X ) −→ T ∗ (S X ) is a complete lattice isomorphism and since

∗ ∗    i = i = i∗ i∈I

i∈I

i∈I

arbitrary infs in the complete lattice T ∗ (S X ) are given by intersection. Now we consider the following categories. • Top, the well-known category of topological spaces and continuous functions.  • L-Top, the category of L-topological spaces (X, ), where  ⊆ L X is a -complete sublattice of the complete lattice L X (L any complete lattice), whose morphisms, namely L-continuous functions f : (X, ) −→ (Y, ), are maps f from X to Y whose backward L-powerset operator f ← : L Y −→ L X reduces to f ← : → , which becomes, of course, a frame map, i.e. an SFrm-morphism. • TopSys, the category whose objects are the topological systems, which are triples (S, M, E) with S any set, M any complete lattice, E ⊆ S × M a binary relation, called the satisfaction relation, such that for all elements of S

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the following equivalences are true ◦ sEa, sEb ⇔ sEa ∧ b; ◦ sE i∈I ai ⇔ ∃i ∈ I : sEai . The morphisms of TopSys((S, M, E), (T, N , D)) are pairs ( f, ) with f ∈ Set(S, T ) and ∈ SLoc(M, N ) such that the following condition is satisfied: f (x)Db ⇔ xE op (b), ∀x ∈ X , b ∈ N where op : N → M is the frame map determining the morphism . Remark 4.1. Topological systems were introduced in [30] and also considered in [5] in connection with L-topological spaces. In both cases the second term of the triple is assumed to be a locale and is therefore a morphism of locales. The possibility of considering these notions in the context of the category of semilocales, as we have done here, is also suggested in [5]. A lattice-valued version of topological systems and its relationships with categories of lattice-valued topological spaces is a matter of recent researches (see [4,29]). Now, for any function f : X → Y , consider f ∗ : S X −→ SY , x  f (x) Then the diagrams relating the powerset operators of f ∗ and the L-powerset operators of f LX ∗

 P(S X )

f L→

/ LY ∗

 / → P(SY )

( f ∗)

LX o

f L←

∗

 P(S X ) o

LY ∗

( f ∗ )←

 P(SY )

are commutative; in fact we have: Proposition 4.2. For any attachment A the following hold 1. ( f L→ (A))∗ = ( f ∗ )→ ( A∗ ); 2. ( f L← (B))∗ = ( f ∗ )← (B ∗ ); 3. f ∗ (x )AB ⇔ x A f L← (B).  Proof. 1.  y ∈ ( f L→ ( A))∗ ⇐⇒  y A f L→ ( A) ⇐⇒ ( f L→ ( A))(y) ∈ F ⇐⇒ {A(x)| f (x) = y} ∈ F ⇐⇒ ∃ x ∈ X such that f (x) = y and A(x) ∈ F ⇐⇒ ∃x ∈ X such that f (x) = y and x ∈ A∗ ⇐⇒ ∃x ∈ A∗ such that f ∗ (x ) =  f (x) =  y ⇐⇒  y ∈ ( f ∗ )→ ( A∗ ). 2. x ∈ ( f L← (B))∗ ⇐⇒ x A f L← (B) ⇐⇒ ( f L← (B))(x) ∈ F ⇐⇒ B( f (x)) ∈ F ⇐⇒  f (x) AB ⇐⇒ f ∗ (x ) ∈ ∗ B ⇐⇒ x ∈ ( f ∗ )← (B ∗ ). 3. f ∗ (x )AB ⇐⇒ f ∗ (x ) ∈ B ∗ ⇐⇒ x ∈ ( f ∗ )← (B ∗ ) ⇐⇒ x ∈ ( f L← (B))∗ ⇐⇒ x A f L← (B).  Proposition 4.3. Let f : X → Y be any function. Then the implication f ∈ L-Top((X ,),(Y , )) ⇒ f ∗ ∈ Top((S X ,∗ ),(SY , ∗ )) is true. Moreover the converse implication is true if the attachment is spatial. Proof. Since by the continuity of f the implication B ∈ ⇒ f L← (B) ∈  is true, it follows from Proposition 4.2 above that B ∗ ∈ ∗ ⇒ ( f ∗ )← (B ∗ ) = ( f L← (B))∗ ∈ ∗ , hence f ∗ is continuous. For the converse implication we recall that under the assumption of spatiality the function * can be reduced to a bijective frame map ∗ :  → ∗ whose inverse *−1 is a bijective frame map too. Then, if f ∗ is continuous, by the

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commutative diagram above f L← can be reduced to the frame map f L← = ∗−1 ◦ ( f ∗ )← ◦ ∗ : −→  so f is L-continuous.  Using obvious notation, it is clear that (g ◦ f )∗ = g ∗ ◦ f ∗ and (i X )∗ = i S X Hence, for any attachment in L the correspondences (X ,)  (S X ,∗ ) f ∈ L-Top((X ,),(Y , ))  f ∗ ∈ Top((S X ,∗ ),(SY , ∗ )) define a functor ∗ : L-Top −→ Top Moreover the following holds. Proposition 4.4. If the attachment is spatial, then the functor ∗ : L-Top → Top is an embedding. Proof. Consider f ∈ L-Top((X, ), (Y, )), g ∈ L-Top((X  ,  ), (Y  ,  )) and assume f ∗ = g ∗ . So, (S X , ∗ ) = (S X  , ∗ ) and (SY , ∗ ) = (SY  , ∗ ), then X = X  , Y = Y  and ∗ = ∗ , ∗ = ∗ . Since (∗)→ is injective,  =  and  = . Besides, ∀  ⊥, ∀x ∈ X ,  f (x) = f ∗ (x ) = g ∗ (x ) = g(x) , and so, f = g.  In [30] the author considered two functors, E V : Top −→ TopSys (X ,O)  (X ,O, ∈) f : (X ,O) → (Y ,U)  ( f ,( f ← )op ) : (X ,O, ∈) → (Y ,U, ∈) and Spat : TopSys −→ Top (S,M,E)  (S,Ext→ (M)) ( f ,g) : (S,M,E) → (T ,N ,D) f : (S,Ext→ (M)) → (T ,Ext→ (N )) where Ext : M → P(S) is the function defined by aExt(a) = {x ∈ S|xEa}. Both fixed-basis and variable-basis functorial relationships between TopSys and L-Top are considered in [5]. Here we show how any attachment on any fixed complete lattice naturally provides a functor ∗ Sys : L-Top −→ TopSys which maps an L-topological space on a set X to a topological system on the set S X of the L-points on X. This functor relates well with E V , Spat and *. We need the following result. Proposition 4.5. For any attachment A one has the following 1. if (X, ) is an L-topological space, then (S X , , A) is a topological system; 2. if f : (X, ) → (Y, ) is L-continuous, then ( f ∗ , ( f L← )op ) : (S X , , A) → (SY , , A) is a morphism of topological systems.  Proof. 1. Follows easily from Lemma 3.8 since  is a -complete sublattice of L X , hence a complete lattice. 2. From the proof of Proposition 4.3 we know that f L← ∈ SFrm( , ). Then the assertion follows by Proposition 4.2, 3. 

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As a consequence the correspondences (X ,)(S X ,,A), f ( f ∗ ,( f L← )op ) define a functor ∗ Sys : L-Top −→ TopSys It can be seen that *Sys is an embedding for every attachment A. Also it is not difficult to see that the triangle ∗ Sys

/ TopSys L-TopG GG vv GG v v G v G v ∗ GG v # zvv Spat Top is commutative, hence * can be deduced by *Sys and Spat, while the triangle ∗ Sys / TopSys L-TopG : GG vv GG v v G v ∗ GGG vv # vv E V Top

is commutative up to isomorphism if the attachment A is spatial. In fact a natural transformation  : E V ◦ ∗ → ∗ Sys can be defined by (X ,) (1 S X ,∗op ) E V ◦ ∗(X ,) −→ ∗ Sys (X ,) = (S X ,∗ , ∈) −→ (S X ,,A)

which becomes a natural isomorphism under the assumption of spatiality of L. Though it is not our main interest, we note that the functors defined here can be extended from the fixed basis to the variable basis context of L-topological spaces. 5. Generalizations and the hypergraph functors It is also possible to describe an attachment on a lattice L with primes as a function A : L ⊥ −→ P(L) In fact, as already remarked, the completely coprime filters are exactly the complements of the principal prime ideals, so the set of completely coprime filters in L can be naturally identified with P(L). Then the function ∗ : L X −→ P(S X ) can be defined by A∗ = {x |A(x) ⱕ A()} Such a description highlights quite well similarities and differences between the functor ∗ : L-Top −→ Top we have discussed here and the hypergraph functor SL considered in different forms by Lowen [22], Rodabaugh [24,25], Gerla [6], Kotzé and Kubiak [19], Höhle [13] (in fact one could speak of hypergraph functors). We use the notation above to describe these functors briefly and to compare those with *. Essentially, they make use of an identity function, say I, instead of A; more precisely I is either the function L  or L ⊥ in [6,22,24,25] or the function P(L) in [13,19]. Note that P(L) coincides with L  in case L is a complete chain, as in [6,22]. Then they define a function from L X to P(X × L  ), where L  may be either L ⊥ or L  or P(L), in one of the following ways: A∗ = {(x,)| A(x) > } A∗ = {(x,)| A(x) ≥ } A∗ = {(x,)| A(x) ⱕ } to construct the hypergraph functor from L-Top to Top. A comparison of the hypergraph functors and the functor ∗ : L-Top → Top would reveal both similarities and differences. We are not interested in giving details of such a

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comparison, nevertheless we remark that the results obtained by means of the hypergraph functors depend only on the lattice structure of L, while our approach makes use of an additional tool (the function A) that in some sense enriches the same structure. In some cases, though, even for the functor * the results depend only on the intrinsic lattice structure of L (see the examples in Section 3). Moreover, note that we share with the work of Kotzé and Kubiak the fundamental use of prime elements of a quite general lattice L. So, in spite of the evident differences in the two approaches, some of our results correspond exactly to results in [19]—compare in particular Propositions 3.9–3.11, 3.13 with Lemma 4.2, Propositions 4.3, 4.5 and Remark 4.4 in [19]. To provide an overview of the different definitions of the hypergraph functors and their links with attachment we require a few preliminaries. Definition 5.1. Let L, M be complete lattices. A ∧-quasi-semilattice morphism from L to M is a function f : L → M that preserves binary infs, i.e. for all a, b ∈ L f (a ∧ b) = f (a) ∧ f (b) it is a quasi-lattice morphism if furthermore it preserves binary sups, i.e. for all a, b ∈ L f (a ∨ b) = f (a) ∨ f (b) it is a quasi-frame morphism if furthermore it preserves arbitrary sups, i.e. for all S ⊆ L f(



S) =



f → (S)

Definition 5.2. Let L be a complete lattice. A subset F (I) of L is a quasi-filter (quasi-ideal) if it is an upper set (a lower set) closed under binary infs (binary sups). A quasi-ideal (quasi-filter) is prime (coprime) if its complement is a quasi-filter (quasi-ideal): it is principal if it is closed under arbitrary sups (arbitrary infs). A quasi-filter F is completely coprime if for every S ⊆ L the implication  S ∈ F ⇒ S ∩ F  ∅ holds. Definition 5.3. An element a in a lattice L is quasi-prime if the implication x ∧ y ≤ a ⇒ x ≤ a or y ≤ a holds. We note that someone considering lattices that are not necessarily bounded could consider superfluous the attribute “quasi” in most occurrences in Definitions 5.1–5.3. In the bounded case, the above definitions clearly extend the corresponding usual notions so as to include trivial cases; in fact the quasi-morphisms we have considered need not preserve the top element (which is the infimum of the empty subset), quasi-filters and quasi-ideals may be empty (unless they are principal),  is quasi-prime. It is not difficult to see that these notions are strictly related to each other, as in the classical case; in particular (just to mention what is interesting to us) a quasi-filter F in L can be characterized as the ∧-kernel, F = f ← ({}), of a ∧-quasi-semilattice morphism f : L → 2, where 2 = {⊥, }, and F is coprime (completely coprime) if and only if f is a quasi-lattice (a quasi-frame) morphism. The complement of a completely coprime quasi-filter is a principal quasi-ideal whose supremum is a quasi-prime element. L-quasi-points in X are pairs (x, ), x ∈ X ,  ∈ L; they can be identified with L-points for   ⊥ and with crisp L-points for  = ; for the latter two we shall also follow the usual notation x . A unified approach to both the hypergraph functor (in its various versions) and the attachment could be provided by means of a function from L to the set ∧-QSLat(L , 2) where we denote by ∧-QSLat the category of complete lattices with ∧-quasi-semilattice morphisms. It is not the aim of this paper to go deeply into details of this unified, very general approach. Nevertheless we describe a possible generalization of the attachment that brings into the same framework the attachment considered in Sections 3 and 4 and the hypergraph functors considered in [6,22,12,19,25] for complete chains. Of course the only quasi-filter in L that is not a filter is the empty subset, which is completely coprime, too.  is the only quasi-prime element that is not prime. The only quasi-frame map from a complete lattice L to the trivial lattice 2 that is not a frame map has constant value ⊥.

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Denote by QFrm the category whose objects are complete lattices and whose morphisms are quasi-frame maps. Moreover denote by CQF(L) the set of all completely coprime quasi-filters in L and by QP(L) the set of all quasi-prime elements of L. It is easily seen that the correspondences f ∈ QFrm(L,2) ∧ -ker ( f ) = f ← ({}) ∈ CQF(L) and a ∈ Q P(L)Fa = {x ∈ L|x ⱕ a} ∈ CQF(L) are bijections. We identify QFrm(L , 2), CQF(L) and PQ(L) by means of these bijections. Definition 5.4. A generalized attachment on a complete lattice L is a function GA : L −→ CQF(L)   F = GA() Of course, the identifications described above allow equivalent formulations of a generalized attachment GA as a function from L to either QP(L) or QFrm(L , 2). The generalized attachment relation can be obviously defined in a similar way as the attachment relation in Section 3. An L-quasi-point (x, ) is said to be relevant for GA if F is non-empty, i.e. if it is attached to  X . All the concepts and results described in Sections 3 and 4 can be suitably restated replacing S X by the set of all the relevant L-quasi-points on L. Clearly, the relevant L-quasi-points are all the L-points, for any attachment in L. Moreover one can argue easily from the above discussion that the hypergraph functors considered in [22,12,19,25] for complete chains are determined by a suitable generalized attachment whose relevant points are those with a prime value—in fact crisp L-points are excluded! Turning to the notion of attachment we remark that this paper has attempted to show that certain (suitable) filters have immediate and effective application in the theory of L-sets and L-topological spaces. cdeo algebras are quite general structures which arise naturally from suitable extensions of an order relation that allows evaluation of implication, conjunction and negation. We have presented detailed examples based on such algebras which, we hope, provide the reader with an appreciation of an attachment between an L-point x and an L-set A. Roughly speaking, in those examples x is attached to A if x exists and it belongs to A, i.e. the conjunction of the existence value, , of x and the membership degree, A(x), of x in A has a value greater than ⊥. When isotonicity holds for an attachment, it means that the attachment of x to A increases its value not only when A(x) increases but also when  does; more explicitly, if x A A,  ≤  and A(x) ≤ B(x) then x A B. It is important to note that isotonicity holds automatically for the above-mentioned attachments on a cdeo algebra L. It is also significant that symmetry in the above-mentioned attachments does not necessarily require commutativity of the conjunction operation (the product—see Example 3.19). These properties of the attachment relation immediately suggest further possible extensions, including: • the meet operation is not essential; in fact conjunction would work even better. Such a replacement would allow weakening the requirement on primes of L assumed in Example 3.19; irreducibility of ⊥ with respect to ⊗ would be enough. This in turn would lead to an axiomatization of L-topological spaces using conjunction, as in [12,15]. • The notion of an L-valued attachment of x to A can be naturally introduced by setting A(x ,A) =  ⊗ A(x) and more generally the attachment degree of A to B would be  A(x) ⊗ B(x) A(A,B) = x∈X

this extension merits consideration in connection with suitable notions of L-valued filters. The ultimate value of the topics covered here is critically linked to their relationship to traditional Chang–Goguen L-topological spaces. Our results justify and motivate most approaches to fuzzy topology using fuzzy points and explain

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in particular why the quasi-coincidence approach yields results very similar to the classical ones, both from a theoretical point of view and from the application side, as in fuzzy topological dynamical systems. Finally, it is sometimes said that analogy is inappropriate as a basis for extending topological concepts to the fuzzy setting. It should be noted, then, that such criticism has been anticipated and discussed in [6] and is the primary motivation for [9]. Acknowledgments Thanks are due to the referees for their helpful comments and remarks. References [1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190. [2] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici, Algebraic Foundation of Many-Valued Reasoning, Trends in Logic, Vol. 7, Kluwer Academic Publishers, Boston, Dordrecht, London, 2000. [3] C. De Mitri, E. Pascali, Characterization of fuzzy topologies from neighbourhoods of fuzzy points, J. Math. Anal. Appl. 93 (1983) 324–327. [4] J.T. Denniston, A. 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