The aim of this paper is to study the (co)completeness of different categories of partial acts. For this, we give constructions of (co)products and (co)equalizers in these categories when possible. While studying cocompleteness, we develop a theory for quotients in these categories as much as possible. We use the same definition of a partial act in every category in question, but we use different kinds of morphisms in these categories.
Hecke–Kiselman monoids were first defined by O. Ganyushkin and V. Mazorchuk in the study of Kiselman quotients of 0-Hecke monoids. The Cycle monoid Cyc_n is the Hecke–Kiselman monoid associated to a cycle consisting of n⩾ 3 vertices. J. Okniński and M. Wiertel gave the reduced form of some elements in the monoid Cyc_n . In this paper, the reduced forms of the remaining elements in Cyc_n are described. Based on these reduced forms, we explicitly construct faithful representations of Cyc_n with and without involution.
The main purpose of this short paper is to present some universal methods to construct a semigroup multiplication on the tensor product L⊗ M of complete lattices L and M such that (L⊗ M, & ) becomes a certain type of algebras. According to isomorphism of this type of algebras, we will investigate the cancellation problem for the tensor product of complete lattices in the category Sup of complete lattices and join-preserving maps.
It is proved that two word problems for ω -terms are decidable: one over the pseudovariety of aperiodic semigroups with commuting idempotents, and the other over the pseudovariety generated by finite aperiodic inverse semigroups. It is shown in particular that these two problems have different solutions. As a consequence, it follows that the above pseudovarieties are different, thereby obtaining an alternative proof for a result of Higgins and Margolis.
We prove that the lattice of varieties of epigroups with completely regular square is arguesian and therefore modular; further, a variety of epigroups with an orthogroup square has a distributive subvariety lattice if and only if its greatest group subvariety possesses the same property. Both the results imply corresponding claims for semigroup varieties. The semigroup analog of the first result was proved in 1991 by Volkov and Ershova but the proof given there contains some inaccuracy. The semigroup analog of the second result was announced by Volkov and Ershova in the early 1990s. In this article we prove both mentioned results in the epigroup setting. In particular, we fix the aforementioned inaccuracy.
