Stojan Bogdanović

Determinization of fuzzy automata with membership values in complete residuated lattices

In this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices. In comparison with the previous methods, developed by Belohlavek [R. Belohlavek, Determinism and fuzzy automata, Information Sciences 143 (2002), 205-209] and Li and Pedrycz [Y.M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005), 68-92], our method always gives a smaller automaton, and in some cases, when the previous methods result in infinite automata, our method can result in a finite one. We also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them, and in particular, to the concept of the Nerodes fuzzy right congruence of a fuzzy automaton, which we introduce and study here.

Myhill--Nerode type theory for fuzzy languages and automata

The Myhill-Nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid. In this paper we develop a general Myhill-Nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1, which are needed to take crisp languages in consideration. We establish connections between extensionality of fuzzy languages w.r.t. right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids, and we prove the Myhill-Nerode type theorem for fuzzy languages. We also prove that each fuzzy language possess a minimal deterministic automaton recognizing it, we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers. In the second part of the paper we introduce and study Nerodes and Myhills automata assigned to a fuzzy automaton with membership values in a complete residuated lattice. The obtained results establish nice relationships between fuzzy languages, fuzzy automata and deterministic automata.

Fuzzy equivalence relations and their equivalence classes

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them.

Uniform fuzzy relations and fuzzy functions

In this paper we introduce and study the concepts of a uniform fuzzy relation and a (partially) uniform F-function. We give various characterizations and constructions of uniform fuzzy relations and uniform F-functions, we show that the usual composition of fuzzy relations is not convenient for F-functions, so we introduce another kind of composition, and we establish a mutual correspondence between uniform F-functions and fuzzy equivalences. We also give some applications of uniform fuzzy relations in approximate reasoning, especially in fuzzy control, and we show that uniform fuzzy relations are closely related to the defuzzification problem.

Fuzzy homomorphisms of algebras

In this paper we consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. In particular, we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations, called uniform fuzzy relational morphisms, and those which are partially uniform F-functions, called fuzzy homomorphisms. Both uniform fuzzy relations and partially uniform F-functions were introduced in a recent paper by us. Uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras. We give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms, we establish certain relationships between them and fuzzy congruences, and we prove homomorphism and isomorphism theorems concerning them. We also point to some applications of uniform fuzzy relational morphisms.

Decompositions of semigroups induced by identities

In this paper we consider decompositions of semigroups induced by identities. Here we give some new characterizations of a semilattice of Archimedean semigroups and, using this, we describe all identities which induce decompositions into a semilattice of Archimedean semigroups. Also, we give a solution for one problem ofШеврин andСуханов [27].

Semilattices of Archimedean semigroups

M. S. Putcha, in 1973, gave the first complete description of semilattices of Archimedean semigroups. Other characterizations of this class of semigroups have been given by T. Tamura in 1972, by S. Bogdanovic and M. Ciric in 1992 and by M. Ciric and S. Bogdanovic in 1993. In this paper, for m, n ∈ N, we define a relation ρ(m,n) and prove that it is a congruence relation on an arbitrary semigroup. Using the congruence relation ρ(m,n) we will give some new characterizations of semilattices of Archimedean semigroups. At the end we describe hereditary properties of semilattices of k-Archimedean semigroups.

Orthogonal sums of semigroups

The purpose of this paper is to prove that every semigroup with the zero is an orthogonal sum of orthogonal indecomposable semigroups. We prove that the set of all 0-consistent ideals of an arbitrary semigroup with the zero forms a complete atomic Boolean algebra whose atoms are summands in the greatest orthogonal decomposition of this semigroup.

Unary algebras, semigroups and congruences on free semigroups

In the triangle consisting of automata, languages and semigroups various correspondences of Eilenbergs type between languages and semigroups and between automata and languages are known, and it remains to establish similar connections between automata and semigroups. In this paper we consider a more general case by taking unary X-algebras instead of automata and we establish complete lattice isomorphisms between the lattices of σ-varieties of X-algebras, κ-varieties of semigroups and weakly invariant congruences on the free semigroup X+, where κ is the cardinality of X, between the lattices of generalized σ-varieties of X-algebras, generalized κ-varieties of semigroups and filters of the lattice of weakly invariant congruences on X+, and between the lattices of pseudo-σ-varieties of X-algebras and pseudo-κ-varieties of semigroups.

Semilattice Decompositions of Semigroups Revisited

Two relations introduced by M. S. Putcha and T. Tamura, denoted by A! and , play a crucial role in semilattice decompositions of semigroups. General properties of the graphs that correspond to these relations were studied by M. S. Putcha in [14], 1974, and the structure of semigroups in which the minimal paths in the graph corresponding to A! are bounded was described by the Þrst two authors in [7], 1996. In this paper this will be done for the relation . We consider the same problem for the left-hand analogue of this relation and for two radicals of the Green’s J -relation. The obtained results generalize various results given by M. S. Putcha, T. Tamura, L. N. Shevrin and the Þrst two authors of this paper.