Congruences on simple ω-semigroups

Abstract

. Introduction and summary. An inverse semigroup whose idempotents form an co-chain e o >e x >e 2 > . . . is called briefly an co-semigroup. A structure theorem for simple <a -semigroups was established by Kocin [7]; a related structure theorem for simple, and also general, to-semigroups was proved by Munn [10]. These results represent an extension of the structure theorem for bisimple a -semigroups due to Reilly [14].Congruences on a bisimple co-semigroup were studied by Munn and Reilly [12]; in particular, they constructed all idempotent-separating congruences and the quotient relative to the least group congruence on such a semigroup. Group congruences on a bisimple co-semigroup were characterized by Ault [1]. Conditions for modularity of the lattice of all congruences on a bisimple co-semigroup were given by Munn [9]. Based on the work of Munn, Baird [2] established conditions for modularity of the lattice of all idempotent-separating and group congruences on a simple co-semigroup. In another communication, Baird [3] described all congruences on a simple co-semigroup contained in a v 2if. Kocin [8] announced certain properties of congruences on a simple co-semigroup as well as of the corresponding homomorphic images.We determine here all congruences on a simple w -semigroup by using a representation for it which is essentially that of Kocin [7]. Determination of congruences throughout this paper is based on their description in [13].By sections, this work is divided as follows. Section 2 contains a minimum of preliminaries concerning simple co-semigroups and congruences on inverse semigroups. The content of Section 3 is a determination of (the lattice of) all normal congruences on the semilattice of idempotents of a simple co-semigroup. A description of normal subsemigroups of a simple co-semigroup contained in the centralizer of idempotents is the subject of Section 4. The results of these two sections make it possible to determine all non-group congruences on a simple co-semigroup; group congruences are characterized in Section 5 by reducing this case to that of a bisimple co-semigroup.We find here an expression for all congruences on a simple co-semigroup. In order to keep the length of the paper within reasonable bounds, we refrain from considering the lattice of congruences, that is, we omit discussing inclusion relations, meets, joins, etc. Neither do we discuss the representations of homomorphic images induced by these congruences. Congruences on non-simple co-semigroups can be deduced from our results in a standard manner.

Modularity of the lattice of congruences of a regular ω-semigroup

“…The congruences on this interesting class of inverse semigroups have been described (see, e.g., [2] and [9]), but this description has been not much used for the investigation of the congruence lattice L(S).…”

“…A regular co-semigroup S is a regular semigroup whose set of idempotents E(S), or shortly E, forms an co-chain under the natural order defined on E by the rule e^.f if and only if ef=f = fe. The congruences on this interesting class of inverse semigroups have been described (see, e.g., [2] and [9]), but this description has been not much used for the investigation of the congruence lattice L(S).…”

Semimodularity of the congruence lattice on regular ?-semigroups

“…The congruences on this interesting class of inverse semigroups have been described (see, e.g., [1], [2] and [11]), but this description has been only recently used for the investigation of the congruence lattice L(S). Therefore several authors characterized bisimple ~o-semigroups whose lattice of congruences satisfies some conditions.…”