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AbstractA semiring is a set $S$with two binary operations $+ $and $\cdot $such that both the additive reduct ${S}_{+ } $and the multiplicative reduct ${S}_{\bullet } $are semigroups which satisfy the distributive laws. If $R$is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B262(1966), 274–277], ${R}_{\bullet } $is a union of groups if and only if ${R}_{\bullet } $is an inverse semigroup if and only if ${R}_{\bullet } $is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum23(1981), 119–136], it is proved that if $R$is a regular ring then ${R}_{\bullet } $is orthodox if and only if ${R}_{\bullet } $is a union of groups if and only if ${R}_{\bullet } $is an inverse semigroup if and only if ${R}_{\bullet } $is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$with ${S}_{+ } $a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum23(1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$for which ${S}_{+ } $is a semilattice.

Abstract: 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.

In this paper we consider the variety UFgenerated 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 UFis 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.

Abstract.: The p n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p n (S) ≤cn r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p n -sequence and give some examples.