Siniša Crvenković

The variety of Kleene algebras with conversion is not finitely based

Given an arbitrary set A, one obtains the full Kleene algebra of binary relations over A by considering the operations of union, composition, reflexive-transitive closure, conversion, and the empty set and the identity relation as constants. Such algebras generate the variety of Kleene algebras (with conversion). As a result of a general analysis of identities satisfied by varieties having an involution operation, we prove that the variety of Kleene algebras with conversion has no finite equational axiomatization. In our argument we make use of the fact that the variety of Kleene algebras without conversion is not finitely based and that, relatively to this variety, the variety of Kleene algebras with conversion is finitely axiomatized.

Finite semigroups with few term operations

Abstract For a semigroup S , p n ( S ) denotes the number of n -ary term operations of S depending on all their variables. The purpose of this paper is to study finite semigroups S with the property that their p n -sequence p ( S )=〈 p 0 ( S ), p 1 ( S ),…〉 is bounded. Such semigroups are described first in terms of identities and then structurally as nilpotent extensions of semilattices, Boolean groups and rectangular bands. As a corollary it is shown that if p ( S ) is bounded then eventually either p n ( S )=0 or 1. It is also shown that there is an effective procedure which decides whether the p n -sequence of a given finite semigroup is bounded or not.

Log-linear varieties of semigroups

A necessary and sufficient condition for a variety of semigroups to be log-linear is found in terms of identities. In particular, every log-linear variety of semigroups is hereditarily finitely based. Also, it can be effectively decided whether a finite semigroup generates a log-linear variety.

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.

On Equations for Union-Free Regular Languages

In this paper we consider the variety UF generated by all algebras of binary relations equipped with the operations of composition, reflexive-transitive closure, and the empty set and the identity relation as constants. This variety coincides with the variety generated by the union-free reducts of Kleene algebras of languages and its free objects are formed by union-free regular languages, that is, regular languages represented by regular expressions having no occurrence of +. We show that the variety UF is not finitely based. The situation does not change if we consider the variety UF? generated by the above algebras of binary relations equipped with the conversion operation.

A note on equations for commutative regular languages

In this note we prove that the equations satisfied by one-letter regular languages are exactly those satisfied by commutative regular languages. This answers a problem raised by Arto Salomaa.

Semigroups with apartness

Proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea used a constructive semigroup with inequality in [8]. This motivated us to achieve a little progress in that direction. The starting point is the structure called a semigroup with apartness. Our primary objective is to prove isomorphism theorems for such constructive semigroups. In doing so our main ideas and notions come from [10].

Congruences on *-Regular Semigroups

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.

A Variety with Undecidable Equational Theory and Solvable Word Problem

In this paper, we exhibit an example of a variety with only one binary operation having recursive base, undecidable equational theory and solvable word problem.

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.