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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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-Lattices

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Elijah Edeghagba Eghosa a , Branimir Šešelja b,∗ , Andreja Tepavˇcevi´c b

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a Bauchi State University, Department of Mathematics, Gadau, Nigeria

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b University of Novi Sad, Faculty of Sciences, Department of Mathematics and Informatics, Trg D. Obradovi´ca 4, 21000 Novi Sad, Serbia

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Received 27 November 2015; received in revised form 14 October 2016; accepted 18 October 2016

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Abstract

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In the framework of -sets, where  is a complete lattice, we introduce -lattices, both as algebraic and as order structures. An -poset is an -set equipped with an -valued order which is antisymmetric with respect to the corresponding -valued equality. Using a cut technique, we prove that the quotient cut-substructures can be naturally ordered. Introducing notions of pseudo-infimum and pseudo-supremum, we obtain a definition of an -lattice as an ordering structure. An -lattice as an algebra is a bi-groupoid equipped with an -valued equality, fulfilling particular lattice-theoretic formulas. On an -lattice we introduce an -valued order, and we prove that particular quotient substructures are classical lattices. Assuming Axiom of Choice, we prove that the two approaches are equivalent. © 2016 Elsevier B.V. All rights reserved.

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* Corresponding author.

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The topic of this research are -valued order structures, where  is a complete lattice. In particular, we deal with -posets and -lattices. Our framework are -sets, introduced 1979. by Fourman and Scott ([15]), whose intention was to model intuitionistic logic. An -set is a nonempty set A equipped with an -valued equality E, where  is a complete Heyting algebra and E is a symmetric and transitive map from A2 to . This notion has been further applied to non-classical predicate logics, and also to foundations of Fuzzy Set Theory ([17,19]). In our approach  is a complete, not necessarily Heyting lattice. The main reason for this co-domain is that it allows main algebraic and set-theoretic properties to be generalized from classical structures to lattice-valued ones,

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1.1. Historical remarks

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1. Introduction

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Keywords: Fuzzy lattice; Fuzzy identity; Fuzzy congruence; Fuzzy equality; Complete lattice

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Research of the 2nd and the 3rd author supported by Ministry of Education and Science, Republic of Serbia, Grant No. 174013.

E-mail addresses: [email protected] (E. Edeghagba Eghosa), [email protected] (B. Šešelja), [email protected] (A. Tepavˇcevi´c). http://dx.doi.org/10.1016/j.fss.2016.10.011 0165-0114/© 2016 Elsevier B.V. All rights reserved.

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E. Edeghagba Eghosa et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

in the framework of cut sets ([22]). So we deal also with lattice-valued structures. These were developed within the Fuzzy Set Theory in which the unit interval has been replaced by a complete lattice (firstly by Goguen [16]). This approach is widely used for dealing with algebraic topics (see e.g., [13], then also [24,25]), and with lattice-valued topology (starting with [20] and many others). In the recent decades a complete lattice is often replaced by a complete residuated lattice ([2]). A lattice-valued equality generalizing the classical one has been introduced in fuzzy mathematics by Höhle in [18], and then it was used in investigations of fuzzy functions and fuzzy algebraic structures by many authors, in particular by Demirci ([10]), Bˇelohlávek and V. Vychodil ([3]) and others. Identities were introduced in fuzzy algebra at the beginning of the century as graded identities, see paper [3] and book [4] by Bˇelohlávek and Vychodil and references therein. Our approach is based on lattice valued identities as given in [26], and then developed in [5–8]. These are lattice-theoretic formulas generalizing classical identities. What is new in this approach is that a lattice-valued identity may hold on an algebra which does not satisfy the analogue classical identity. Due to the topic of our research, we shortly comment the development of lattice-valued ordering structures. In [2] an L-order with an L-equality relation is introduced as a fuzzy relation satisfying reflexivity, antisymmetry w.r.t. L- equality and transitivity. It is different from -poset that is used in our approach in the type of reflexivity we use, which makes the framework completely different. In [30], a lattice-valued (fuzzy) lattice was introduced both, as a lattice valued set on a lattice, and as a special lattice-valued poset. Two definitions were proved to be equivalent, as in the classical case. There were also several further approaches to fuzzy lattices ([1]). Some of these were connected to fuzzy formal concept analysis (see [2]). Let us also mention some recent investigations of fuzzy (complete) lattices ([21,31]). In particular, in a series of papers ([10–12]), Demirci investigates fuzzy equality, and in this framework fuzzy functions and fuzzy lattices as ordering structures. Recently, -algebras, in particular -groups, were introduced in [8,28]. In this context, an approach to -lattice appeared in [27]. Our last comment here concerns the notion of reflexivity of a binary lattice valued relation R on a set A. It is usually defined by R(x, x) = 1, where 1 is the top of the membership values lattice. Our approach follows another direction (for the fuzzy approach see e.g., [19,29], and for the crisp motivation, [14,15,23]). Namely, we consider R to be a lattice valued relation on a lattice valued subset μ of A, meaning that the membership values R(x, y) could not be greater then either of μ(x), μ(y). Therefore, if R is reflexive on μ, then R(x, x) = μ(x). The crisp motivation of this approach is evaluated in Preliminaries, along with the definition of μ-reflexivity.

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1.2. Results of the paper

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As mentioned, we use a structure (M, E) called -set, consisting of a nonempty set M and an -valued equality E on M;  denotes a complete lattice. An -poset (M, E, R) is an -set (M, E) equipped with an appropriate -order R. An -lattice is an -poset, in which there are so called pseudo-suprema and pseudo-infima for twoelement subsets. Using cut sets, we prove that particular quotient substructures of an -lattice are classical lattices. Next we introduce an -lattice as an algebra. This is a structure (M, E) where M = (M, , ) is a bi-groupoid, (M, E) is an -set, and E is compatible with operations in M. In addition, special lattice-theoretic formulas are supposed to be satisfied. We prove basic lattice-like properties of this structure. Completing these investigations, we prove that, under the assumption of the Axiom of Choice, the two notions are equivalent. Namely, it is possible to define operations on an -lattice as an ordered structure, so that it becomes an -lattice as an algebra. On the other hand on an -lattice as an algebra, it is possible to define an -valued order, under which this structure is an -lattice. We also prove that the classical notions of lattice valued ordered structures are special cases of the analogue notions introduced here. Let us mention that some aspects of this research initially appeared in the paper [27], still most of the results are firstly introduced and developed here.

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1.3. Organization of the paper

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In the preliminary section we review algebraic and lattice theoretic notions, and we briefly present relevant part of the theory of -valued functions, -sets and -algebras. The subsequent section contains the results of the paper, divided in subsections. First we introduce the notion of an -poset and prove that the corresponding -valued order induces a natural order on particular cut quotient structures. Next we define the -lattice, requiring the existence of so-called pseudo-infimum and pseudo-supremum for every two-element subset. We prove that the cut quotient posets are classical lattices. Further, starting with an -set whose domain is equipped with two binary operations (hence being a bi-groupoid), we define the -lattice as an algebra. Namely, in order that such structure fulfills lattice identities (commutativity, associativity and absorption laws for the two binary operations), special lattice-theoretic formulas should hold. We prove some main properties of the new structure, like idempotent laws. These are supposed to be valid also for the bi-groupoid itself, provided that the -valued equality fulfills the so called separation property. We also introduce an -valued order on an -lattice as an algebra. By particular antisymmetry, this order is related to the -valued equality on the structure. Finally, we prove the equivalence of two -lattice notions. For this purpose we use the Axiom of Choice. Namely, we prove that on an -lattice as an -poset, it is possible to define two binary operations so that this structure becomes an -lattice as an algebra. Conversely, an -lattice as an algebra can be equipped with a naturally defined -valued order, under which this structure is an -lattice as an ordered structure. We illustrate our research by a suitable example.

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2.1. Lattices

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Here we deal with a generalization of a lattice as a poset and as an algebraic structure. In our approach we use particular functions whose co-domain is a fixed complete lattice. So, we consider a lattice to be an ordered structure (L, ) in which for every two-element subset {a, b} there exist the infimum (greatest lower bound) denoted by a ∧ b and the supremum (least upper bound), denoted by a ∨ b. The lattice is complete if for every subset there exist the infimum and the supremum. A complete lattice possesses the smallest (the bottom) and the greatest (the top) elements, 0 and 1, respectively. Among other known notions related to lattices, we use a principal filter in L, generated by an element p ∈ L, denoted by ↑p: ↑p := {x ∈ L | p  x}.

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As it is known, an equivalent notion to a lattice as a poset is the lattice as an algebra (algebraic structure) (L, ∧, ∨) with two binary operations ∧ and ∨, both commutative, associative and fulfilling the absorption laws ((x ∧ y) ∨ x = x and (x ∨ y) ∧ x = x, for all x, y ∈ L). These and other related notions can be found in every textbook about order structures, see e.g., [9]. In the paper we use the following version of the Axiom of Choice: (AC) For a collection X of nonempty subsets of a set M, there exists a function f : X → M, such that for every A ∈ X , f (A) ∈ A.

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2.2. -Valued functions and relations

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2. Preliminaries

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Throughout the paper, (, ∧, ∨, ) is a complete lattice with the top and the bottom elements 1 and 0 respectively. An -valued function μ on a nonempty set A is a mapping μ : A → . For p ∈ L, a cut set or a p-cut of an -valued function μ : A →  is a subset μp of A which is the inverse image of the principal filter in , generated by p:

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μp = μ−1 (↑p) = {x ∈ X | μ(x)  p}.

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An -valued (binary) relation R on A is an -valued function on A2 , i.e., it is a mapping R : A2 → .

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R is symmetric if R(x, y) = R(y, x) for all x, y ∈ A;

(1)

R is transitive if R(x, y)  R(x, z) ∧ R(z, y) for all x, y, z ∈ A.

(2)

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Let μ : A →  and R : A2 →  be an -valued function and an -valued relation on A, respectively. Then we say that R is an -valued relation on μ if for all x, y ∈ A R(x, y)  μ(x) ∧ μ(y).

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(x, y) ∈ R imply x, y ∈ μ.

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An -valued relation R on μ : A →  is said to be reflexive on μ or μ-reflexive, if

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(4)

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(x, x) ∈ R if and only if x ∈ μ,

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which means that a relation R on A is reflexive on a subset μ of A. A symmetric and transitive -valued relation R on A, which is reflexive on μ : A →  is an -valued equality on μ. As above, in the bivalent case a symmetric and transitive relation R on A is an equivalence relation on a subset μ of A: μ = {x ∈ A | (x, x) ∈ R}

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R(x, y)  R(x, x) ∧ R(y, y) (strictness) .

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(5)

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If for all x, y ∈ A R(x, y) = R(y, x) = R(x, x) = R(y, y) implies x = y (separation),

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then R is said to be a separated -valued relation on A. An -set is a pair (A, E), where A is a nonempty set, and E is an -valued equality over A, which is defined as a symmetric and transitive -valued relation on A. The following is straightforward.

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Proposition 1. An -valued equality E on a set A fulfills the strictness property.

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For an -set (A, E), we denote by μ the -valued function on A, defined by μ(x) := E(x, x).

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We say that μ is determined by E. Clearly, by strictness property, E is an -valued relation on μ, namely, it is an -valued equality on μ. In terms of the fuzzy set theory, μ is a fuzzy subset of A, and E is a fuzzy equality on μ. Lemma 1. If (A, E) is an -set and p ∈ , then the cut Ep is an equivalence relation on the corresponding cut μp of μ.

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Let R →  be an -valued relation over a nonempty set A. R is said to fulfill the strictness property if for all x, y ∈ A

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Again, in the crisp version, formula (4) becomes

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Observe that if  is a two-element chain, then μ is a (characteristic function of a) subset of A, and (3) means

R(x, x) = μ(x) for every x ∈ A.

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Proof. Reflexivity of Ep over μp : (x, x) ∈ Ep if and only if E(x, x) = μ(x)  p, if and only if x ∈ μp . Symmetry and transitivity are proved straightforwardly. 2

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2.4. -Algebra; identities

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If A = (A, F ) is an algebra, then as it is known, a lattice-valued subalgebra of A, in our terms an -valued subalgebra of A is a mapping μ : A →  which is not constantly equal to 0, and which fulfills the following: For any operation f from F with arity greater than 0, f : An → A, n ∈ N, and for all a1 , . . . , an ∈ A, we have that n 

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μ(ai )  μ(f (a1 , . . . , an )),

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μ(ai )  μ(t (a1 , . . . , an )).

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Let A = (A, F ) be an algebra. An -valued relation R : →  is compatible with the operations in F if the following holds: for every n-ary operation f ∈ F and for all a1 , . . . , an , b1 , . . . , bn ∈ A A2

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R(c, c) = 1 for every constant (nullary operation) c ∈ F.

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Next we introduce a notion of a lattice-valued algebra with a lattice valued equality. Let A = (A, F ) be an algebra and E : A2 →  an -valued equality on A, which is compatible with the operations in F . Then we say that (A, E) is an -algebra. The following facts about -algebras are direct consequences of some well known properties of lattice valued (fuzzy) structures and their cuts. Proposition 3. Let (A, E) be an -algebra. Then the following hold:

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(i) The function μ : A →  determined by E (μ(x) = E(x, x) for all x ∈ A), is an -valued subalgebra of A. (ii) For every p ∈ , the cut μp of μ is a subalgebra of A, and (iii) For every p ∈ , the cut Ep of E is a congruence relation on μp .

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Proof. (i) Let f ∈ F be an n-ary operation on A. Then for all for all x1 , . . . , xn ∈ A

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E(xi , xi )  E(f (x1 , . . . , xn ), f (x1 , . . . , xn )) = μ(f (x1 , . . . , xn ))

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by the compatibility of E. (ii) Straightforward, by the definition of a cut. (iii) We prove that Ep is a congruence relation on μp . It is an equivalence relation on μp by Lemma 1. In addition, Ep is compatible with the operations in μp . Indeed, take f ∈ F , x1 , . . . , xn , y1 , . . . , yn ∈ μp , and suppose that for i = 1, . . . , n, (xi , yi ) ∈ Ep . Then for every i, E(xi , yi )  p. Now, we have that

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E(f (x1 , . . . , xn ), f (y1 , . . . , yn )) 

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Proposition 2. Let μ : A →  be an -valued subalgebra of an algebra A and let t (x1 , . . . , xn ) be a term in the language of A. If a1 , . . . , an ∈ A, then the following holds:

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and for a nullary operation (constant) c ∈ F , μ(c) = 1. The following is a lattice-valued version of a known property of term operations in universal algebra (see [8]).

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i.e., (f (x1 , . . . , xn ), f (y1 , . . . , yn )) ∈ Ep .

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Now we deal with identities which hold on -algebras, adopting the approach from [26]. Let (A, E) be an -algebra and u(x1 , . . . , xn ) ≈ v(x1 , . . . , xn ) (briefly u ≈ v) an identity in the language of A, where variables appearing in terms u and v are among x1 , . . . , xn Then, (A, E) satisfies identity u ≈ v (i.e., this identity holds on (A, E)) if the following condition is fulfilled for all a1 , . . . , an ∈ A and for the term-operations uA and v A on A corresponding to terms u and v respectively: n 

μ(ai )  E(u (a1 , . . . , an ), v (a1 , . . . , an )). A

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Proposition 4. If an identity u ≈ v holds on an algebra A, then this identity also holds on an -algebra (A, E).

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3. -Lattice

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3.1. -Poset; -lattice as ordered structure

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Let E be an -valued equality on a nonempty set A. We say that an -valued relation R : A2 →  on A is E-antisymmetric, if the following holds: R(x, y) ∧ R(y, x) = E(x, y), for all x, y ∈ A.

(12)

Let (M, E) be an -set. We say that an -valued relation R : M 2 →  on M is an -valued order on (M, E), if it fulfills the strictness property (5), it is E-antisymmetric, and it is transitive in the sense of (2). A structure (M, E, R) is an -poset, if (M, E) is an -set, and R : M 2 →  is an -valued order on (M, E). In addition, it is clear that by (12), R(x, x) = E(x, x), for every x ∈ M. As indicated by (7), we denote by μ the -valued function on M, defined by μ(x) = E(x, x). Obviously, both E and R are reflexive relations on μ, in the sense of (4). By Lemma 1, every cut Ep of E is a classical equivalence relation on the cut μp of μ. In this context, as usual, we denote by [x]Ep the equivalence class of x ∈ μp , and by μp /Ep the corresponding quotient set: for p ∈ 

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[x]Ep := {y ∈ μp | xEp y}, x ∈ μp ; μp /Ep := {[x]Ep | x ∈ μp }.

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Proposition 5. Let (M, E, R) be an -poset. Then for every p ∈ , the binary relation ≤p on μp /Ep , defined by [x]Ep ≤p [y]Ep if and only if (x, y) ∈ Rp

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is a classic ordering relation. Proof. First, we prove that the relation ≤p is well defined. Indeed, suppose that [x]Ep ≤p [y]Ep , i.e., that (x, y) ∈ Rp . Now, if u ∈ [x]Ep , v ∈ [y]Ep , then E(x, u) = R(x, u) ∧ R(u, x)  p and similarly E(y, v) = R(y, v) ∧ R(v, y)  p. Thereby, since by assumption R(x, y)  p, using transitivity of R we obtain p  R(x, u) ∧ R(u, x) ∧ R(y, v) ∧ R(v, y) ∧ R(x, y)  R(u, v) ∧ R(x, u) ∧ R(v, y)  R(u, v). Therefore, [u]Ep ≤p [v]Ep and the order ≤p does not depend on class representatives. The relation ≤p is reflexive on μp /Ep : for x ∈ μp , [x]Ep ≤p [x]Ep if and only if (x, x) ∈ Rp if and only if R(x, x)  p if and only if E(x, x)  p if and only if μ(x)  p if and only if x ∈ μp . Next, ≤p is antisymmetric: Suppose that for x, y ∈ μp , [x]Ep ≤p [y]Ep and [y]Ep ≤p [x]Ep . This is equivalent with (x, y) ∈ Rp and (y, x) ∈ Rp , which holds if and only if R(x, y) ∧ R(y, x) = E(x, y)  p, i.e., if and only if [x]Ep = [y]Ep .

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Transitivity: Let [x]Ep ≤p [y]Ep and [y]Ep ≤p [z]Ep , i.e., (x, y) ∈ Rp and (y, z) ∈ Rp , hence equivalently R(x, y)  p and R(y, z)  p, if and only if R(x, y) ∧ R(y, z)  p. Then, by transitivity of R, R(x, z)  p, i.e., (x, z) ∈ Rp , which finally gives [x]Ep ≤p [z]Ep . 2

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μ(a) ∧ μ(b)  E(c, c1 ) if and only if c1 is also a pseudo-infimum of a and b. Analogously, if d is a pseudo-supremum of a and b, then μ(a) ∧ μ(b)  E(d, d1 ) if and only if d1 is also a pseudo-supremum of a and b.

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μ(a) ∧ μ(b)  R(c, a) ∧ R(c, b) ∧ R(c1 , a) ∧ R(c1 , b)  R(c1 , c) ∧ R(c, c1 ) = E(c, c1 ). Conversely, let μ(a) ∧ μ(b)  E(c, c1 ). Then, for every p  μ(a) ∧ μ(b), p  μ(a) ∧ μ(b)  R(c, c1 ) ∧ R(c1 , c), hence p  R(c1 , c). Since p  R(c, a) ∧ R(c, b), we have p  R(c1 , c) ∧ R(c, a) ∧ R(c1 , c) ∧ R(c, b)  R(c1 , a) ∧ R(c1 , b). In addition, assume x ∈ μp and p  R(x, a) ∧ R(x, b). Then, p  R(x, c). Since also p  μ(a) ∧ μ(b)  R(c, c1 ), by transitivity we obtain p  R(x, c1 ). The proof for pseudo-suprema is analogous. 2

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Remark 2. Since for p  q, every equivalence class of μq /Eq is contained in a class of μp /Ep , we get that pseudo-infima (suprema) of two elements a, b, if they exist, belong to the same equivalence class in μp /Ep , for p  μ(a) ∧ μ(b).

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Proof. Suppose that c and c1 are pseudo-infima of a, b ∈ M. Then by (i),

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Proposition 6. Let (M, E, R) be an -poset and a, b, c, c1 , d, d1 ∈ M. If c is a pseudo-infimum of a and b, then

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Remark 1. It is straightforward that a pseudo-infimum (supremum) of a and b belongs to μp for every p  μ(a) ∧ μ(b). Observe that a pseudo-infimum and a pseudo-supremum for given a, b ∈ M, if they exist, are not unique in general. In the following proposition we prove that pseudo-infima (suprema) of two elements a, b, if they exist, they belong to the same equivalence class in μp /Ep , for p  μ(a) ∧ μ(b).

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(ii) μ(a) ∧ μ(b)  R(a, d) ∧ R(b, d) and for every p  μ(a) ∧ μ(b) for every x ∈ μp p  R(a, x) ∧ R(b, x) implies p  R(d, x).

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An element d ∈ M is a pseudo-supremum of a, b ∈ M, if the following holds:

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(i) μ(a) ∧ μ(b)  R(c, a) ∧ R(c, b) and for every p  μ(a) ∧ μ(b), for every x ∈ μp p  R(x, a) ∧ R(x, b) implies p  R(x, c).

10 11

2

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Let (M, E, R) be an -poset and a, b ∈ M. An element c ∈ M is a pseudo-infimum of a and b, if the following holds:

7 8

1

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Remark 3. By the above definition, if E is a separated equality on M, for p = μ(a), the unique pseudo-infimum (supremum) of one element a ∈ M (i.e., for a and b with a = b), is a. In terms of Proposition 6, this follows from the fact that for every a ∈ M, the class [a]Eμ(a) consists of the single element a: x ∈ [a]Eμ(a) if and only if E(a, x)  μ(a) if and only if E(a, x)  E(a, a) if and only if x = a.

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We say that an -poset (M, E, R) is an -lattice as an ordered structure, if for every a, b ∈ M there exist a pseudo-infimum and a pseudo-supremum. In the following, infimum and supremum of elements a and b in an ordered set (here a lattice) are denoted by inf(a, b) and sup(a, b), respectively.

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Theorem 1. Let (M, E, R) be an -poset. Then it is an -lattice as an ordered structure if and only if for every q ∈ , the poset (μq /Eq , ≤q ) is a lattice, where the relation ≤q on the quotient set μq /Eq is defined by (13) and the following holds: for all a, b ∈ M, and p = μ(a) ∧ μ(b), inf([a]Ep , [b]Ep ) ⊆ inf([a]Eq , [b]Eq ) and sup([a]Ep , [b]Ep ) ⊆ sup([a]Eq , [b]Eq ), for every q, q  p.

Proof. Let (M, E, R) be an -poset. Then, by Proposition 5, for every q ∈ , (μq /Eq , ≤q ) is a classical poset. Assume now than in addition, (M, E, R) is an -lattice. Under this assumption (μq /Eq , ≤q ) is a (classical) lattice. Indeed, let a, b ∈ μq , and let c be a pseudo-infimum of a, b. Then, q  μ(a) ∧ μ(b), and also c ∈ μq by the definition of a pseudo-infimum (see also Remark 1). Hence by (13), [c]Eq ≤q [a]Eq and [c]Eq ≤q [b]Eq , and for every x ∈ μq , if [x]Eq ≤q [a]Eq and [x]Eq ≤q [b]Eq , then also [x]Eq ≤q [c]Eq . Therefore, [c]Eq = inf([a]Eq , [b]Eq ) in μq /Eq . By the above, (14) holds for infima, since q  p, for p = μ(a) ∧ μ(b), and therefore inf([a]Ep , [b]Ep ) = [c]Ep ⊆ [c]Eq = inf([a]Eq , [b]Eq ). Analogously, using (ii), we can prove that the class [d]Eq is a supremum of [a]Eq and [b]Eq in μq /Eq , where d is a pseudo-supremum of a and b. Similarly as for infima, (14) holds for suprema. Hence, for every q ∈ , (μq /Eq , ≤q ) is a lattice and (14) holds. Conversely, suppose that for a given -poset (M, E, R), the poset (μp /Ep , ≤p ) is a lattice for every p ∈ , and that (14) holds. We prove that then for all a, b ∈ M there exist a pseudo-infimum and a pseudo-supremum. Indeed, since for p = μ(a) ∧μ(b), (μp /Ep , ≤p ) is a lattice, there is the infimum for the classes [a]Ep and [b]Ep , a class [c]Ep , for some c ∈ μp . Hence [c]Ep ≤p [a]Ep , [c]Ep ≤p [b]Ep , and for every x ∈ μp , if [x]Ep ≤p [a]Ep , [x]Ep ≤p [b]Ep , then also [x]Ep ≤p [c]Ep . By Proposition 5, this is equivalent with (c, a), (c, b) ∈ Rp , and (x, a), (x, b) ∈ Rp implies (x, c) ∈ Rp . By (14), the above properties of c hold if p is replaced by q, q  p. Therefore, c is a pseudo-infimum of a and b. Analogously one can prove that there is a pseudo-supremum for a and b. 2

41 42 43 44 45 46 47 48 49 50 51 52

4

6 7 8

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Remark 4. According to Theorem 1 and Remark 3, if E is a separated equality in an -lattice (M, E, R) and if μ1 = ∅, then (μ1 , R1 ) is a lattice. Clearly, in this case E1 is a diagonal relation and the congruence classes under E1 are one-element sets. Then also R1 is an order on μ1 and the posets (μ1 , R1 ) and (μ1 /E1 , ≤1 ) are order isomorphic. Since the latter is a lattice by Theorem 1, the same holds for the former. Due to the same argument, for every p ∈ , (μp , Rp ) is a lattice whenever equivalence classes under Ep are all one-element sets.

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3.2. -Lattice as -algebra

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(14)

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Here we define an -lattice as an -algebra, according to the definition in Section 2.4. This algebraic approach was firstly developed in [27]. Hence, basic definitions and main results from this paper are also given in the present section. However, we adopt these results to the wider framework of both, algebraic and relational approach. In addition, there is a difference in the definition of the separation property. Namely, as it is defined in [27], for a separated -equality E it is possible that for some x we have E(x, x) = 0. Consequently, an -lattice (generally an -algebra) could be considered to be a proper sublattice (subalgebra) of the basic bi-groupoid (algebra), which is not our intention here: we deal with -sets, -lattices, generally with -algebras, not with their substructures. We start with a bi-groupoid – an algebra M = (M, , ) with two binary operations, without any additional conditions. Next, we assume that E : M 2 →  is an -valued equality on M, hence that (M, E) is an -set. In addition, E should be compatible with operations  and  in the following sense: E(x, y) ∧ E(z, t)  E(x  z, y  t) and E(x, y) ∧ E(z, t)  E(x  z, y  t). The following are straightforward properties of the above notions.

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Proposition 7. If E is a compatible -valued equality on a bi-groupoid M = (M, , ), and μ : M →  is defined by μ(x) = E(x, x), then the following hold:

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μ(x) ∧ μ(y)  μ(x  y) and μ(x) ∧ μ(y)  μ(x  y).

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(15)

(ii) For every p ∈ , the cut μp of μ is a sub-bi-groupoid of M. (iii) For every p ∈ , the cut Ep of E is a congruence relation on μp .

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Proof. (i) for all x, y ∈ M

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μ(x) ∧ μ(y) = E(x, x) ∧ E(y, y)  E(x  y, x  y) = μ(x  y) by the compatibility of E. Similarly we get the second formula in (15). (ii) and (iii) are straightforward consequences of Proposition 3 and the part (i) above.

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We say that the -algebra (M, E) is an -lattice as an -algebra (-lattice as an algebra), if it satisfies lattice identities:

18

1 : x  y ≈ y  x

20

2 : x  y ≈ y  x

3 : x  (y  z) ≈ (x  y)  z

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4 : x  (y  z) ≈ (x  y)  z

25 26 27 28 29

5 : (x  y)  x ≈ x 6 : (x  y)  x ≈ x.

23 24

(absorption)

25 26

As defined by (11), this means that for all x, y, z ∈ M, the following formulas are satisfied, where, as already indicated, the mapping μ : M →  is defined by μ(x) = E(x, x): L2 : μ(x) ∧ μ(y)  E(x  y, y  x)

32 33

L3 : μ(x) ∧ μ(y) ∧ μ(z)  E((x  y)  z, x  (y  z))

34

L4 : μ(x) ∧ μ(y) ∧ μ(z)  E((x  y)  z, x  (y  z))

38 39 40 41

22

(associativity)

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L5 : μ(x) ∧ μ(y)  E((x  y)  x, x) L6 : μ(x) ∧ μ(y)  E((x  y)  x, x).

(commutative laws) (associative laws)

Next we present some additional properties of -lattices as algebras. As mentioned, these propositions (to the end of Sections 3.2) are adopted versions of the results in [27]. We give them here in order to make the whole presentation complete, but we omit proofs. The proof of the following lemma follows by Lemma 1 in [27]. Lemma 2. An -lattice (M, E) fulfills the following versions of the absorption identities:

50 51 52

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36 37 38 39 40 41

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(y  x)  x ≈ x and (y  x)  x ≈ x.

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The next result is proved as Proposition 3 in [27].

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(absorption laws)

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L1 : μ(x) ∧ μ(y)  E(x  y, y  x)

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(commutativity)

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(i) For all x, y ∈ M,

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Proposition 8. In an -lattice (M, E) as an algebra, the idempotent identities x  x ≈ x and x  x ≈ x are valid.

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Under the separation property the idempotency in the -valued framework implies that the same identity should be satisfied by the bi-groupoid on which an -lattice is defined (Proposition 4 in [27]):

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If (M, E) is an -lattice with M being a classical lattice, then, by Proposition 7, μ is an -valued sublattice of M. However, an -lattice is a notion which is more general then an -valued sublattice. Namely, if M is a bi-groupoid which is not a lattice and (M, E) is an -lattice, then obviously μ is by Proposition 7 an -valued sub-bi-groupoid of M, hence it is not an -valued sublattice of M in general. Next we describe the notion of an -lattice (M, E) in the framework of classical lattices obtained by cuts of the bi-groupoid M. Observe that by Proposition 7, for every p ∈ , the cut μp is a sub-bi-groupoid of M and Ep is a congruence relation on μp . Hence, μp /Ep is a quotient bi-groupoid of μp over Ep . The theorem that follows is analogue to Theorem 1 which is dealing with -lattices as ordered structures. The present one is concerned with -lattices as algebras; it is already given in [27], still we present a slightly reformulated proof.

26 27

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Proposition 10. If M = (M, , ) is a lattice and E is a compatible -valued equality on M, then (M, E) is an -lattice.

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If a bi-groupoid M is a classical lattice, and E is an -valued compatible equality on M, then (M, E) is an -lattice, as we prove in the sequel. In other words, classical lattice properties imply formulas L1–L6 (see Proposition 5 in [27]).

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Proposition 9. [27] Let (M, E) be an -lattice, in which E is a separated -valued equality. Then the idempotent law x  x ≈ x is valid in (M, E) if and only if the operation  is idempotent in the bi-groupoid M = (M, , ), and analogously x  x ≈ x holds in (M, E) if and only if  is idempotent in M.

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1

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Theorem 2. Let M = (M, , ) be a bi-groupoid, and let E be an -valued compatible equality on M. Then, (M, E) is an -lattice if and only if for every p ∈ , the quotient structure μp /Ep is a lattice. Proof. Let M = (M, , ) be a bi-groupoid and E a compatible -equality on M, such that (M, E) is an -lattice. According to the above comment, for p ∈ , we consider the quotient set of μp over Ep , denoted as usual by μp /Ep . In addition, we denote operations on congruence classes by p and p . These operations are introduced in a natural way (by class representatives) and it is easy to prove that they are well defined. Now, (μp /Ep , p , p ) is a bi-groupoid. We prove that this bi-groupoid is a lattice. Let [x]Ep , [y]Ep , [z]Ep be elements (classes) from μp /Ep . We have to prove lattice axioms 1–6. We prove 1, commutativity of p : For x, y ∈ μp , since (M, E) is an -lattice, by L1 we have E(x  y, y  x)  μ(x)  μ(y)  p,

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hence (x  y, y  x) ∈ Ep . Therefore, for x, y ∈ μp , we have that [x]Ep p [y]Ep = [x  y]Ep = [y  x]Ep = [y]Ep p [x]Ep , so we proved that the operation p in μp /Ep is commutative. Similarly we can prove the remaining five lattice axioms, hence μp /Ep is a lattice. Conversely, suppose that for every p from , the quotient structure μp /Ep is a lattice. Now, we have to prove that (M, E) is an -lattice, i.e., that formulas L1–L6 hold. We prove the absorption law L6 μ(x) ∧ μ(y)  E((x  y)  x, x),

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the others are proved analogously. Let μ(x) ∧ μ(y) = p. Then x, y ∈ μp . Since (μp /Ep , p , p ) is a lattice by assumption, by 6 we have ([x]Ep p [y]Ep ) p [x]Ep = [x]Ep , hence [(x  y)  x]Ep = [x]Ep , and therefore E((x  y)  x, x)  p = μ(x) ∧ μ(y).

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We conclude the section by describing the relationship between quotient lattices μp /Ep , for various p ∈ .

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Proposition 11. Let (M, E) be an -lattice and p, q ∈ , with p  q. Then, the mapping f : μq /Eq → μp /Ep , defined by f ([x]Eq ) = [x]Ep is a lattice homomorphism.

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

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Proof. The function f is well defined, since by assumption p  q, hence μq ⊆ μp ; therefore, if x ∈ μq , then x ∈ μp . For the same reason, if [x]Eq = [y]Eq , then also [x]Ep = [y]Ep . Now for x, y ∈ μq , we have and analogously for the operation q .

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3.3. Equivalence of two approaches

13

3.3.1. -Lattice as ordered structure is -lattice as algebra Here we prove that, like in the classical case, the notions of an -lattice as an algebraic structure and an -lattice as an ordered structure are equivalent. In the following we assume that the Axiom of Choice (AC) holds. Definition 1. Let (M, E, R) be an -lattice as an ordered structure. We define two binary operations,  and  on M as follows: for every pair a, b of elements from M, a  b is an arbitrary, fixed pseudo-infimum of a and b, and a  b is an arbitrary, fixed pseudo-supremum of a and b. Assuming Axiom of Choice, by which an element is chosen among all pseudo-infima (suprema) of a and b and then this element is fixed, the operations  and  on M are well defined. Indeed, by the definition of an -lattice, for any a, b ∈ M, a pseudo-infimum and a pseudo-supremum exist; by Definition 1, they are unique. If E is a separated -valued equality, then by Remark 3, for every a ∈ M, we get a  a = a and a  a = a, i.e., in this case these operations are idempotent. Hence, the structure M = (M, , ) is a bi-groupoid. In the following proposition we prove that μ (μ(x) = E(x, x)) is an -valued sub-bigroupoid of M. Proposition 12. Let (M, E, R) be an -lattice, μ : M →  defined by (7) (μ(x) = E(x, x)) and M = (M, , ) a bi-groupoid, as defined above. Then, for all x, y ∈ M μ(x) ∧ μ(y)  μ(x  y) and μ(x) ∧ μ(y)  μ(x  y).

6

8

f ([x]Eq q [y]Eq ) = f ([x  y]Eq ) = [x  y]Ep = [x]Ep p [y]Ep ,

(16)

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Proof. Let x, y ∈ M and x  y = z. Denote μ(x) ∧ μ(y) by p. Then, by Remark 1, the class [z]Ep exists, i.e., z ∈ μp . By the definition of a cut, we have μ(z)  μ(x) ∧ μ(y). The proof in case z = x  y is analogue. 2

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4 5

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3

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Next, we deal with an -lattice (M, E, R) as an ordered structure, in which, by Theorem 1, for every p ∈ , the quotient structure (μp /Ep , ≤p ) is a lattice, where ≤p is defined by (13) and (14) holds. For x, y ∈ μp , we denote infimum and supremum of [x]Ep and [y]Ep by [x]Ep p [y]Ep and [x]Ep p [y]Ep , respectively.

40

Lemma 3. Let (M, E, R) be an -lattice as an ordered structure, and p ∈ . Then for all x, y ∈ μp , in the lattice (μp /Ep , ≤p ),

44

[x]Ep p [y]Ep = [x  y]Ep and [x]Ep p [y]Ep = [x  y]Ep ,

41 42 43

45 46 47

where  and  are the operations on M introduced by Definition 1.

48

Proof. By Proposition 5, in the lattice (μp /Ep , ≤p ), relation ≤p is an ordering relation given by

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[x]Ep ≤p [y]Ep if and only if (x, y) ∈ Rp .

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By the definition of a pseudo-infimum, since x, y ∈ μp and p  μ(x) ∧ μ(y), we get (x  y, x) ∈ Rp , (x  y, y) ∈ Rp , hence [x  y]Ep ≤p [x]Ep and [x  y]Ep ≤p [y]Ep . Therefore, [x  y]Ep is a lower bound for [x]Ep and [y]Ep . Further, if for some u ∈ μp we have (u, x) ∈ Rp and (u, y) ∈ Rp , then also by the definition of a pseudo-infimum, (u, x  y) ∈ Rp . Equivalently, if [u]Ep ≤p [x]Ep and [u]Ep ≤p [y]Ep , then [u]Ep ≤p [x  y]Ep . Hence [x  y]Ep is the greatest lower bound (infimum) for [x]Ep and [y]Ep . The proof that [x  y]Ep is the supremum for [x]Ep and [y]Ep , i.e., that [x]Ep p [y]Ep = [x  y]Ep is analogue. 2

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16 17 18 19 20 21 22 23 24 25

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E(x, y) ∧ E(u, v)  E(x  u, y  v), and analogously for the operation . By the definition of a cut, the above inequality is equivalent with (x  u, y  v) ∈ Ep , where p = E(x, y) ∧ E(u, v). Since E(x, y)  p and E(u, v)  p, i.e., (x, y), (u, v) ∈ Ep , our task is to prove that Ep is a congruence relation on the sub-bigroupoid μp of M. Indeed, if (x, y), (u, v) ∈ Ep , then [x]Ep = [y]Ep and [u]Ep = [v]Ep in the lattice μp /Ep . Then, [x]Ep p [u]Ep = [y]Ep p [v]Ep , implying by Lemma 3, [x  u]Ep = [y  v]Ep , which gives (x  u, y  v) ∈ Ep . Compatibility with  is proved analogously. 2

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15 16 17 18 19 20 21 22 23 24 25

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Proposition 14. Let (M, E, R) be an -lattice as an ordered structure, and ,  the corresponding binary operations on M, introduced by Definition 1. Then, the formulas L1–L6 are satisfied.

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26

Now we prove that the operations  and  satisfy lattice-theoretic identities 1, . . . , 6, which means that the formulas L1, . . . , L6 hold.

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4

14

Proof. We have to prove that for all x, y, u, v ∈ M,

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Proposition 13. Let (M, E, R) be an -lattice as an ordered structure, and ,  the corresponding binary operations on M, introduced by Definition 1. Then, E is compatible with  and .

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As we proved in Proposition 12, the function μ : M →  (μ(x) = E(x, x)) is an -valued sub-bigroupoid of (M, , ). Therefore, for every p ∈ , μp is a (classical) sub-bigroupoid of (M, , ). We use this in the following proposition.

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1

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Proof. We prove that the formula L1, commutativity of , holds: Since for every p ∈ L, μp /Ep is a lattice, we have that for all x, y ∈ M, with μ(x) ∧ μ(y) = p, [x]Ep p [y]Ep = [y]Ep p [x]Ep . Therefore [x  y]Ep = [y  x]Ep ,

33 34 35 36 37 38 39

40

and thus, by the definition of these classes,

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E(x  y, y  x)  p = μ(x) ∧ μ(y).

41

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Using the same (analogue) argument, we prove that all the remaining five formulas hold. 2

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Proof. This is a straightforward consequence of Propositions 12, 13 and 14 and of the definition of -lattice as an algebra. 2

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Theorem 3. If (M, E, R) is an -lattice as an ordered structure, and M = (M, , ) the bi-groupoid in which operations ,  are introduced in Definition 1, then (M, E) is an -lattice as an algebra.

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Finally, let us prove a property of -lattices as ordered structures which is analogue to the following known fact about lattices: x ≤ y if and only if x ∧ y = x.

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Proposition 15. If (M, E, R) is an -lattice as an ordered structure, then for all x, y ∈ M,

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μ(x) ∧ μ(y) ∧ E(x  y, x) = R(x, y).

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Proof. For every p ∈ , (μp /Ep , ≤p ) is a lattice, hence for all x, y ∈ M, by Proposition 5 and by strictness property of R, we have R(x, y)  p if and only if x, y ∈ μp and (x, y) ∈ Rp if and only if μ(x) ∧ μ(y)  p and [x]Ep ≤p [y]Ep if and only if μ(x) ∧ μ(y)  p and [x]Ep p [y]Ep = [x]Ep if and only if μ(x) ∧ μ(y)  p and [x  y]Ep = [x]Ep if and only if μ(x) ∧ μ(y) ∧ E(x  y, x)  p. So, for every p ∈  and for all x, y ∈ M μ(x) ∧ μ(y) ∧ E(x  y, x)  p if and only if R(x, y)  p,

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which proves the proposition. 2

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3.3.2. -Lattice as algebra is -lattice as ordered structure Here we deal with the relational aspect of -lattices as algebras. First we introduce an -valued order on these structures. Theorem 4. Let M = (M, , ) be a bi-groupoid and (M, E) an -lattice as an algebra, where E is a separated -valued equality on M. Then the -valued relation R : M 2 → , defined by R(x, y) := μ(x) ∧ μ(y) ∧ E(x  y, x)

(17)

12 13 14 15 16 17 18 19 20

is an -valued order on M.

21 22

Proof. We have to prove that R is E-antisymmetric and transitive. R is E-antisymmetric:

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25

E(x, y) = E(x, y) ∧ E(y, x) ∧ E(x, x) ∧ E(y, y) 

25

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E(x  y, x  x) ∧ E(y  x, y  y) ∧ E(x, x) ∧ E(y, y) =

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E(x  y, x) ∧ E(y  x, y) ∧ E(x, x) ∧ E(y, y) = R(x, y) ∧ R(y, x) =

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E(x  y, x) ∧ E(y  x, y) ∧ E(x, x) ∧ E(y, y) 

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E(x, x  y) ∧ E(x  y, y  x) ∧ E(y  x, y) 

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E(x, y  x) ∧ E(y  x, y)  E(x, y),

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by Proposition 1 (formula (5)), by compatibility of E; we also use idempotency of  (Proposition 9, since E is separated by assumption), axiom L1 and transitivity of E. Hence,

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R(x, y) ∧ R(y, x) = E(x, y),

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proving that R is E-antisymmetric. R is transitive:

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R(x, y) ∧ R(y, z) =

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E(x, x) ∧ E(y, y) ∧ E(x  y, x) ∧ E(z, z) ∧ E(y  z, y) 

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E((x  y)  z, x  z) ∧ E(x  (y  z), x  y) ∧ E(x, x) ∧ E(y, y) ∧ E(x  y, x) ∧ E(z, z) 

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E((x  y)  z, x  z) ∧ E(x  (y  z), x  y) ∧ E(x  (y  z), (x  y)  z)∧

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E(x, x) ∧ E(x  y, x) ∧ E(z, z) 

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E(x  y, x  z) ∧ E(x  y, x) ∧ E(x, x) ∧ E(z, z) 

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E(x  z, x) ∧ E(x, x) ∧ E(z, z) = R(x, z).

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Observe that in case E is separated, the diagonal part of R coincides with the corresponding sub-relation of E: for all x ∈ M R(x, x) = E(x, x).

(18)

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Fig. 1. Membership values lattice.

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Indeed, by Proposition 8, R(x, x) = E(x  x, x) = E(x, x). We also prove that the relation R on an -lattice determines the order on the cut-lattices. Proposition 16. Let M = (M, , ) be a bi-groupoid, (M, E) an -lattice as an algebra, and R : M 2 →  an -valued relation on M defined by (17). Let p ∈ . Then, for x, y ∈ μp and [x]Ep , [y]Ep ∈ μp /Ep ,

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[x]Ep ≤p [y]Ep if and only if xRp y.

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Proof. By Theorem 2, μp /Ep is a lattice, hence

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[x]Ep ≤p [y]Ep if and only if x, y ∈ μp and [x]Ep p [y]Ep = [x]Ep if and only if x, y ∈ μp and [x  y]Ep = [x]Ep

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if and only if x, y ∈ μp and (x  y, x) ∈ Ep

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if and only if μ(x) ∧ μ(y) ∧ E(x  y, x)  p if and only if R(x, y)  p

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if and only if xRp y.

2

Theorem 5. Let M = (M, , ) be a bi-groupoid, (M, E) an -lattice as an algebra in which E is separated, and R : M 2 →  an -valued relation on M defined by R(x, y) := μ(x) ∧ μ(y) ∧ E(x  y, x). Then, (M, E, R) is an -lattice as an ordered structure.

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Proof. Under the given assumptions, by Theorem 4, (M, E, R) is an -poset. Since (M, E) is an -lattice as an algebra, for every p ∈ , the quotient sub-bi-groupoid μp /Ep is a lattice with respect to operations p and p induced on the congruence classes by the operations  and  respectively. By Proposition 16, the order ≤p is induced by R, i.e., it is precisely the order defined by (13). Finally, (14) holds. Indeed, for a, b ∈ M and p  μ(a) ∧ μ(b), we have [a]Ep p [b]Ep = [a  b]Ep ⊆ [a  b]Eq = [a]Eq q [b]Eq . Now by Theorem 1, (M, E, R) is an -lattice as an ordered structure. 2

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3.3.3. Concrete example In this part we present an example of a finite -lattice. It is constructed as an ordered structure (M, E, R), and the operations are introduced by Definition 1. Of course, the two approaches (order-theoretic and algebraic) are equivalent. Let M = {a, b, c, d, e, f, g}, and let  be a membership values lattice given in Fig. 1. An -valued, separated equality E : M 2 →  is given in Table 1, and in Table 2 we present an -valued transitive relation R : M 2 → , which, in addition, satisfies the strictness property; moreover, as it is necessary by the definition, the formula E(x, y) = R(x, y) ∧ R(y, x) holds for all x, y ∈ M. Now (M, E, R) is an -lattice as an ordered structure. This fact is shown by the following analysis. -function μ : M → , defined by μ(x) := E(x, x), is given by

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Table 1 -valued equality E. E a b c a r p 0 b p r 0 0 0 s c d 0 0 q 0 0 q e f 0 0 0 0 0 0 g

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d 0 0 q 1 q 0 0

e 0 0 q q 1 0 0

f 0 0 0 0 0 q 0

g 0 0 0 0 0 0 q

Table 2 -valued order R. R a b c a r r 0 b p r 0 0 0 s c d r r s 0 0 q e f 0 0 0 0 0 0 g

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d 0 0 q 1 q 0 0

e r r s 1 1 0 0

f 0 0 q q q q 0

g 0 0 q q q q q

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Fig. 2. Quotient lattices.

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Table 3 Operation .  a a a b a c d d d e a f d ∗∗ g a ∗∗

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 μ=

a r

b r

c s

b a b d d b a ∗∗ e∗∗

d 1

c d d c d c d∗ c∗

e 1

d d d d d d e∗ e∗

f q

g q

e a b c d e c∗ c∗

f b∗∗ a ∗∗ c∗ d∗ e∗ f f

g c∗∗ g ∗∗ c∗ d∗ c∗ f g

Table 4 Operation .  a a a b b c e a d e e g ∗∗ f g b∗∗

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b b b e b e g ∗∗ g ∗∗

c e e c c e f g

d a b c d e f g

e e e e e e f g

f f ∗∗ a ∗∗ f f f f g

g a ∗∗ c∗∗ g g g g g

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.

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The cuts of μ and the cuts of E represented by partitions are: μ0 = M ; μp = {a, b, c, d, e} ; μq = {c, d, e, f, g} ; μr = {a, b, d, e} ; μs = {c, d, e} ; μt = μ1 = {d, e};

= M 2;

E0 Ep = {{a, b}, {c}, {d}, {e}}; Eq = {{c, d, e}, {f }, {g}}; Er = {{a}, {b}, {d}, {e}}; Es = {{c}, {d}, {e}}; Et = E1 = {{d}, {e}}.

For all x ∈  the quotient structures μx /Ex are lattices with respect to the order defined by (13); μ0/E0 is obviously a one-element lattice, and the other quotient lattices are represented in Fig. 2. Observe that by Remark 4, (μr , Rr ), (μs , Rs ), (μt , Rt ) and (μ1 , R1 ) are also lattices, since the corresponding congruence classes under the cuts of E are one-element sets. Finally, two binary operations on M constructed by means of pseudo-infima and pseudo-suprema, according to Definition 1, are given in Tables 3 and 4. The operations are idempotent, since E is separated. Observe that in some fields of these tables the values could be arbitrary (the values denoted by ∗∗) since the corresponding class is the

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whole set M, or they could be chosen among several elements (c, d, e) belonging to the same class (values indicated by ∗); we give possible choices of these values. In this way, we obtain the bi-groupoid M = (M, , ). By Theorem 3, (M, E) is an -lattice as an algebra.

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4. Conclusion

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-lattices, as introduced here, appear as a continuation of our research of -algebras. They are based on -sets, they are defined as -algebras, and in addition they are extended to -relational structures. An essential aspect of this investigation are identities which are satisfied up to an -equality. Therefore, basic structures are not lattices, but the quotient structures constructed on cuts are. Which could be, in our opinion, the following steps of this research? Firstly, there are many particular lattice and order-theoretic notions whose precise definitions and the role in this -framework are still unclear (e.g., homomorphisms, particular -sublattices, special elements etc.). Next, there are strong reasons (originating in fuzzy logic) for using a (particular) residuated lattice instead of a general complete lattice  (actually, -sets were defined with  being a Heyting algebra). But in this case cut structures would not preserve classical algebraic properties. Therefore, connections to known structures should be established in a different way than in the case when the language uses only classical lattice-theoretic operations. References [1] [2] [3] [4] [5] [6] [7]

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