Rozália Madarász

Algebras of fuzzy sets

In this paper we investigate two kinds of algebras of fuzzy sets, which are obtained by using Zadehs extension principle. We give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets. We prove that if the structure of truth values is a complete residuated lattice, the induced algebra of a subalgebra of an algebra A can be embedded into the induced algebra of fuzzy sets of A. For direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product. In the case of homomorphisms and direct products, the two kinds of algebras of fuzzy sets behave in different ways.

On Kleene algebras

Abstract In this paper we prove that the class of inversion-free Kleene algebras is not finitely based. The main idea is to use a result of Redko and Salomaa for regular languages. We also prove unsolvability of the word problem for Kleene algebras and some other varieties of algebras.

Construction of finite L-groups

Abstract In this paper we give some necessary and sufficient conditions for a fuzzy relation to be a compatible fuzzy equality of some L-group in the sense of Bělohlavek. These characterizations enable practical applications for construction of L-groups with a given skeleton. The construction has two steps: determination of the general form of fuzzy pre-equalities on a group and then solving a system of inequalities in the given residuated lattice. Some concrete examples of construction of finite L-groups with the given skeleton are presented.

Correspondence: Remarks on the lattices of fuzzy subsets of a groupoid

In this paper we give a necessary and sufficient condition for a groupoid D such that the sup-min product is distributive over arbitrary intersection of fuzzy subsets of D, and correct some results from the paper [S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239-245]. Also, we prove that the set of all idempotent fuzzy sets forms a complete lattice, which is a complete join-sublattice of the lattice of all fuzzy subgroupoids. This result extends the corresponding result from the above mentioned paper.

On dynamic algebras

Abstract Dynamic algebras are the Lindenbaum–Tarski algebras of dynamic logics. These algebras can be considered as Boolean algebras with some operators, indexed by the elements of some Kleene algebra. In this paper we prove that there are infinitely many finitely generated varieties of dynamic algebras having undecidable equational theories. All these varieties are generated by representable dynamic algebras.

Some Globally Determined Classes of Graphs

For a class of graphs we say that it is globally determined if any two nonisomorphic graphs from that class have nonisomorphic globals. We will prove that the class of so called CCB graphs and the class of finite forests are globally determined.

Compatibility of fuzzy power relations

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.

On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties.

Strong retracts of unary algebras

This paper introduces the notion of a strong retract of an algebra and then focuses on strong retracts of unary algebras. We characterize subuniverses of a unary algebra which are carriers of its strong retracts. This characterization enables us to describe the poset of strong retracts of a unary algebra under inclusion. Since this poset is not necessarily a lattice, we give a necessary and sufficient condition for the poset to be a lattice, as well as the full description of the poset.