On regular semigroups

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe . If E = {e i : i∈ N }, where N denotes the set of all non-negative integers, and if the elements of E form the chain then S is called an ω- semigroup .

Regular ω-semigroups

If S is a (multiplicative) Semigroup and a ∈ S, the binary operation ∘ defined on the set S by x ∘ y = x a y is associative and the resulting semigroup is called a variant of S. We study the congruence a defined on S by saying that two elements are α-related if and only if they determine the same variant of S. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involving D-equivalence).

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On variants of a semigroup

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

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Enlargements of regular semigroups

A class of contracted inverse semigroup rings

A congruence ρ on a semigroup is said to be idempotent-separating if each ρ-class contains at most one idempotent. For any idempotent e of a semigroup S the set eSe is a subsemigroup of S with identity e and group of units H e , the maximal subgroup of S containing e . The purpose of the present note is to show that if S is a regular O-bisimple semigroup and e is a non-zero idempotent of 5 then there is a one-to-one correspondence between the idempotentseparating congruences on 5 and the subgroups N of H e with the property that aN ⊆ Na for all right units a of eSe and Nb ⊆ bN for all left units b of eSe. Some special cases of this result are discussed and, in the final section, an application is made to the principal factors of the full transformation semigroup on a set X.

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W. D. Munn

Papers

The Idempotent-separating Congruences on a Regular 0-bisimple Semigroup