Structure of coextensions of regular semigroups by rectangular bands

In this paper, we described the structure of coextensions of regular semigroups by rectangular bands.     

Some open problems in the structure theory of regular semigroups

The aim of this note is to summarise the most important results which generalise those describing the structure of inverse semigroups via semidirect products of semilattices by groups, and to formulate several open problems whose investigation might contribute to the development of this topic. This is the written version of the author's talk given at the problem session of the Colloquium on Semigroups, Szeged, 2000. The paper also contains some problems raised there by D. B. McAlister and P. A. Grillet. December 7, 2000     

Completely regular semigroups

No abstract. August 22, 2001     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.     

Finitely approximable regular semigroups

In this note it is proved that a regular semigroup whose subgroups are all finitely approximable is finitely approximable and that the set of idempotents of each principal factor is finite. As a corollary necessary and sufficient conditions are found for certain classes of regular semigroups to be finitely approximable.Translated from Matematicheskie Zametki, Vol. 17, No. 3, pp. 423–432, March, 1975.The author is grateful to L. N. Shevrin and Yu. N. Mukhin for their valuable observations and helpful discussions.     

Completely regular semigroups as semigroups of partial transformations

Communicated by G. Lallement     

Relatively congruence-free regular semigroups

Yu, Wang, Wu and Ye call a semigroup \(S\) \(\tau \) -congruence-free, where \(\tau \) is an equivalence relation on \(S\) , if any congruence \(\rho \) on \(S\) is either disjoint from \(\tau \) or contains \(\tau \) . A congruence-free semigroup is then just an \(\omega \) -congruence-free semigroup, where \(\omega \) is the universal relation. They determined the completely regular semigroups that are \(\tau \) -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is \(\mathrel {\mathcal {J}}\) -congruence-free if and only if it is either a semilattice or has a single nontrivial \(\mathrel {\mathcal {J}}\) -class, \(J\) , say, and either \(J\) is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for \(\mathrel {\mathcal {L}}\) and \(\mathrel {\mathcal {R}}\) . In the case of \(\mathrel {\mathcal {H}}\) , only the completely semisimple case is fully resolved, again specializing to those of Yu et al.     

Congruences on regular semigroups

The kernel of a congruence on a regular semigroup S may be characterized as a set of subsets of S which satisfy the Teissier-Vagner-Preston conditions. A simple construction of the unique congruence associated with such a set is obtained. A more useful characterization of the kernel of a congruence on an orthodox semigroup (a regular semigroup whose idempotents form a subsemigroup) is provided, and the minimal group congruence on an orthodox semigroup is determined.     

Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

Subdirect products of regular semigroups

A nonempty subset X contained in anH-class of a regular semigroup S is called agroup coset in S if XX′X=X and X′XX′=X′ where X′ is the set of inverses of elements of X contained in anH-class of S. Let μ denote the maximum idempotent separating congruence on S. We show in Section 1 of this paper that the set K(S) of group cosets in S contained in the μ-classes of S is a regular semigroup with a suitably defined product. In Section 2, we describe subdirect products of twoinductive groupoids in terms of certain maps called ‘subhomomorphisms’. A special class of subdirect products, called S^*-direct products, is described in Section 3. In the remaining two sections, we give some applications of the construction of S^*-direct products for describing coextensions of regular semigroups and for providing a covering theorem for pseudo-inverse semigroups.     

Congruences on eventually regular semigroups

Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored.