Narrow Results

A residuated lattice is an ordered algebraic structure

The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒ^C that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

The structure of simple and directly indecomposable members of the variety RICRL of representable idempotent commutative residuated lattices is described. These descriptions are employed to determine the size and structure of free algebras in RICRL, using existing theory on the Fraser–Horn and Apple properties. A formula for the free spectrum of RICRL is given.

In this paper we extend some previously established links between the derivation operators used in formal concept analysis and some mathematical morphology operators to fuzzy concept analysis. We also propose to use mathematical morphology to navigate in a fuzzy concept lattice and perform operations on it. Links with other lattice-based for malisms such as rough sets and F-transforms are also established. This paper proposes a discussion and new results on such links and their potential interest.

In this paper, the lattice operations and the adjoint pair on the fuzzy filters set on residuated lattices are defined, the conclusion that the fuzzy filters lattice defined as such is a distributive residuated lattice is obtained. An order-reversing involution on the fuzzy strong-prime filters sublattice is introduced. It is proved that the fuzzy strong-prime filters sublattice is a quasi-Boolean algebra.

In this paper, the concept of intuitionistic fuzzy sets is applied to residuated lattices. The notion of intuitionistic fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of intuitionistic fuzzy filters are obtained. We show that the set of all the intuitionistic fuzzy filters of a residuated lattice forms a complete lattice and we find the distributive sublattices of it. Finally, the correspondence theorem for intuitionistic fuzzy filters is established.

The concept of recognizability of a fuzzy language by crisp deterministic fuzzy automaton and monoid is well known. The purpose of the present work is to study the recognizability of fuzzy languages by fuzzy ordered monoids. We show that the upper sets of a given fuzzy ordered monoid play a nice role in such studies. Also, we introduce the syntactic fuzzy ordered monoid of an upper set which recognizes this upper set.

In this paper, we introduce the concepts of $n$-fold boolean ideals, $n$-fold implicative ideals and $n$-fold integral ideals in residuated lattices and we state and prove their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of $n$-fold ideal and $n$-fold filter in residuated lattices.

Given a complete residuated lattice ${ℒ}:=(L;\wedge ,\vee ,\ominus ,\twoheadrightarrow ,⊸;0,1)$ and a mono-unary algebra ${𝒜}:=(A;f)$, it is well known that ${ℒ}$ and the residuated lattice ${ℱ}u(A,L):=(\mathrm{\text{Fu}}(A,L);\wedge ,\vee ,\ominus ,\twoheadrightarrow ,⊸;\underline{0},\underline{1})$ of $L$-fuzzy subsets of $A$ satisfy the same residuated lattice identities. In this paper, we show that ${ℒ}$ and the residuated lattice ${ℱ}s({𝒜},L):=(\mathrm{\text{Fs}}({𝒜},L);\wedge ,\vee ,\ominus ,\hookrightarrow ,\looparrowright ;\underline{0},\underline{1})$ of $L$-fuzzy subalgebras of ${𝒜}$ satisfy the same residuated lattice identities if and only if the Heyting algebra ${𝒮}ub({𝒜}):=(\mathrm{\text{Sub}}({𝒜});\cap ,\cup ,\Rightarrow ;\varnothing ,A)$ of subuniverses of ${𝒜}$ is a Boolean algebra. We also show that ${ℱ}s({𝒜},L)$ is a Boolean algebra (respectively, an $MV$-algebra) if and only if ${ℒ}$ is a Boolean algebra (respectively, an $MV$-algebra) and ${𝒮}ub({𝒜})$ is a Boolean algebra.

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

In this paper, we introduce soaker filters in a residuated lattice, give some characterizations and investigate some properties of them. Then we define a topology on the set of all the soaker filters, which we call reflectional topology, show it is an Alexandrov topology and give a basis for it. We introduce the notion of join-soaker filters and prove that when the residuated lattice is a join-soaker filter, then the reflectional topology is compact. We also give a characterization of connectedness of the reflectional topology. Finally, we prove the reflectional topology is $T_0$, give necessary and sufficient conditions under which it is $T_1$ and prove that being $T_2$ is equivalent to being $T_1$. Several illustrative examples are given.

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

The study examines the categorical connections of $L$-G-filters with $L$-filters and $L$-interior operators. Besides, the relationship between $L$-G-filters and $L$-fuzzy pre-proximities is also explored. Indeed, the Galois correspondence between the categories of stratified $L$-G-filter spaces and $L$-fuzzy pre-proximity spaces is revealed in this paper.

In this paper, under a residuated lattice, the concepts of left-orthogonal and right-orthogonal in inner product spaces are introduced and their connection with fuzzy Galois connections are established.

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

Presented is an algorithm for constructing MRAP-sequences from sets of fuzzy attribute implications (FAIs). FAIs are formulas used to describe dependencies in object-attribute data with graded attributes as well as functional dependencies in ranked data tables over similarity relations. MRAP-sequences are particular normalized proofs from sets of FAIs. Using MRAP-sequences, one can discover dependencies following from other dependencies which is an important task in relational data analysis. We present theoretical foundations and the algorithm. All proofs are omited due to the limited scope of this paper.

In general, the F-transform is a powerful technique to approximate functions or compress data. In this article we explore the residuated versions of F-transform and extend their results for data that can not be represented by a function. We present some properties and demonstrate their applicability on an illustrative example.

This paper introduces an idea, how the so-called excluding symptoms could be incorporated into the fuzzy relational compositions and provides an overview of valid properties.

Rough sets, a tool for data mining, deal with the vagueness and granularity in information systems. This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. Topological properties of L-fuzzy rough sets model are discussed and that a map between rough set model and a infinite set can induce a L-fuzzy similar relation on the infinite set is pointed out.

The aim of this paper is to develop further the fuzzy filter theory of general residuated lattices. Mainly, we introduce the concept of (∈, ∈ ∨_q)-fuzzy regular filters, and derive some characterizations of them. Moreover, we discuss some relations among several special (∈, ∈ ∨_q)-fuzzy filters.

Bosbach and Riečan states on residuated lattices both are analogues of probability measures on Boolean algebras. In this paper it is proved that Riečan states on a residuated lattice is uniquely determined by its restriction on the subset of all regular elements and consequently the same holds for Bosbach states on Glivenko residuated lattices. These results indicate that for studies of Riečan and Bosbach states one can concentrate only on involutive residuated lattices.