Search

The purpose of this paper is to investigate topics in fuzzy concept analysis using the theory of arrow and fuzzy categories. Our approach deals with the original, fuzzy data using fuzzy subsets in order to obtain a notion of fuzzy concepts, fuzzy concept lattices, and attribute implications between fuzzy sets of attributes. Our approach is general in the sense that we only require a Heyting algebra together with a t-norm like operation as the lattice of truth values instead of the unit interval [ 0 , 1 ] or a substructure thereof. As a consequence, regular concept analysis, i.e., concept analysis based on regular sets and the Boolean truth values is just a special case of our approach. In addition, our approach differs from a lot of approaches in the literature, which usually first apply some kind of defuzzyfication, e.g., α -cuts, and then apply concept analysis in the classical (crisp) sense. [ABSTRACT FROM AUTHOR]

An algebra 〈 A , ∧ , ∨ , → , □ , 0 , 1 〉 of type (2 , 2 , 2 , 1 , 0 , 0) is said to be a modal weak Heyting algebra if 〈 A , ∧ , ∨ , → , 0 , 1 〉 is a weak Heyting algebra and the following conditions are satisfied for every a , b ∈ A : M1) □ (1) = 1 , M2) □ (a ∧ b) = □ (a) ∧ □ (b) and M3) □ (a → b) ≤ □ (a) → □ (b). If this algebra satisfies the inequality a ∧ (a → b) ≤ b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members. [ABSTRACT FROM AUTHOR]

In order to facilitate the processing of vast amounts of data that emerges in the study of fuzzy social networks, scholars have developed various procedures for reducing these networks. Some procedures use regular and structural fuzzy relations to reduce such networks. In this article, we generalize the notions of regular and structural fuzzy relations to obtain even better reductions of fuzzy networks. More precisely, for a fuzzy social network given by the set of social entities and the family of fuzzy relations between them, we define μ -approximate regular fuzzy relations, where μ is the degree taken from the underlying set of truth values, which is a complete Heyting algebra. Using these specific fuzzy relations, we show that it is possible to reduce a fuzzy social network in some cases when previously developed algorithms fail to reduce it. We investigate the properties of μ -approximate regular fuzzy relations. We show that the blockmodel of a fuzzy social network, which is its reduced fuzzy social network built via μ -approximate regular fuzzy preorder, retains specific structure-preserving properties. We give a method for calculating the greatest μ -approximate regular fuzzy relation on a given fuzzy social network. For fuzzy social networks defined over the real-unit interval [ 0 , 1 ] , we give a procedure that determines all subintervals of [ 0 , 1 ] that share the greatest μ -approximate regular fuzzy relation. Analogous results are provided for μ -approximate structural fuzzy relations. [ABSTRACT FROM AUTHOR]

In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras. [ABSTRACT FROM AUTHOR]

Simplification logic, a logic for attribute implications, was originally defined for Boolean sets. It was extended to distributive fuzzy sets by using a complete dual Heyting algebra. In this paper, we weaken this restriction in the sense that we prove that it is possible to define a simplification logic on fuzzy sets in which the membership value structure is not necessarily distributive. For this purpose, we replace the structure of the complete dual Heyting algebra by the so-called weak complete dual Heyting algebra. We demonstrate the soundness and completeness of this simplification logic, and provide a characterisation of the operations defining weak complete dual Heyting algebras. [ABSTRACT FROM AUTHOR]

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L -fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces. [ABSTRACT FROM AUTHOR]

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. In this paper, we extend monadic MV-algebras with a state operator that describes algebraic properties of states. The resulting variety of algebras will be called state monadic MV-algebras. First, we introduce state monadic MV-algebras and establish a natural equivalence between the category of state monadic MV-algebras and the category of state monadic ℓ -groups with strong units. Moreover, we prove that the class of all state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. In particular, by studying extended state monadic ideals, we prove that the set of all stable state monadic ideals in state monadic MV-algebras is a complete Heyting algebra. Also, the class of all involutory state monadic ideals in state monadic MV-algebras is a complete Boolean algebra. Finally, we introduce and characterize some members in the variety of state monadic MV-algebras, which are subdirectly irreducible, simple, semisimple, local and semilocal, respectively. [ABSTRACT FROM AUTHOR]

In this paper we give an explicit description of the left adjoint of the forgetful functor from the algebraic category of Gödel algebras (i.e., prelinear Heyting algebras) to the algebraic category of bounded prelinear Hilbert algebras. We apply this result in order to study possible descriptions of the coproduct of two finite algebras in the algebraic category of prelinear Hilbert algebras. [ABSTRACT FROM AUTHOR]

In this paper, notions of left- and right-state operators on residuated lattices are introduced and some related properties of such operators are investigated. Filters and normal filters generated by a subset in a state residuated lattice are characterized and it is shown that the lattice of filters forms a frame. Subdirectly irreducible state residuated lattices are characterized. The notion of state coannihilator is introduced and a connection between them and Galois connection is established. Finally, it is shown that the set of state coannihilators forms a complete Boolean algebra. [ABSTRACT FROM AUTHOR]

T -rough Heyting algebras were introduced by Eric SanJuan in 2008 as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. In this paper, we introduce the variety of algebras, which we call T -rough symmetric Heyting algebras. These algebras constitute an extension of T -rough Heyting algebras and a generalization of symmetric Heyting algebras of order n. Our main interest is the representation theory of tense operators on T -rough symmetric Heyting algebras. In order to do this, a discrete-style duality for these algebras is developed. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a prefilter generated by a nonempty subset of an EQ-algebra is introduced and a characterization of it is obtained. It is proved that the set of all prefilters of an EQ-algebra is an algebraic lattice and it is a Brouwerian lattice for an ℓ EQ-algebra. Furthermore, it is shown that the set of all principal prefilters of an ℓ EQ-algebra is a sublattice of the lattice of prefilters. Then by defining an implication between two prefilters, it is determined that the lattice of prefilters is a Heyting algebra, for an ℓ EQ-algebra. Finally, the EQ-algebras for which the lattice of prefilters is a Boolean algebra are given. [ABSTRACT FROM AUTHOR]

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate derivations in residuated lattices and characterize some special types of residuated lattices in terms of derivations. In the paper, we discuss related properties of some particular derivations and give some characterizations of (good) ideal derivations. Then we obtain that the fixed point set of good ideal derivations is still a residuated lattice. Also, we prove that the set of all ideal derivation filters on residuated lattices with good ideal derivations can form a complete Heyting algebra, which is isomorphic to the lattice of all filters in the fixed point set for good ideal derivations. Moreover, we study principal ideal derivations and their adjoint derivations. Finally, we get that the fixed point set of principal ideal derivations and that of their adjoint derivations are order isomorphism. In particular, using the fixed point set of principal ideal derivations, we give characterizations of Heyting algebras and linearly ordered Heyting algebras. [ABSTRACT FROM AUTHOR]

Arrow categories establish a categorical and algebraic description of L -fuzzy relations, i.e., relations that use membership values from an arbitrary but fixed complete Heyting algebra L . With other words arrow categories describe the fixed-basis case. In this paper we are interested in the variable-basis case, i.e., the case where relations between different objects may use different membership values. We will investigate the structure of the collection of lattices of membership values within a given Dedekind category. This will lead to a complete characterization of the variable-basis case in this context. [ABSTRACT FROM AUTHOR]

In this paper we study the conditions under which the compatibility of binary fuzzy relations is preserved by the operation of powering. We pay particular attention to fuzzy power algebras based on the cartesian product of fuzzy sets. We show that the preservation of compatibility essentially depends on the underlying structure of truth values L . Among other things we prove that for fuzzy preorders both Hoare-goodness and Smyth-goodness imply compatibility, but the converse is generally true only under the assumption that L is a complete Heyting algebra. [ABSTRACT FROM AUTHOR]