Abundant Semigroups with Adequate Transversals

Abstract

Suppose that S is a regular semigroup and S an inverse subsemigroup of S. S is called an inverse transversal of S, if for each a ∈ S, S meets V (a) precisely once (that is, |V (a) ∩ So| = 1). The structure theorems for regular semigroups with inverse transversals have been given by many authors (see [1, 2, 9, 10]). An analogue of an inverse transversal, which is termed an an adequate transversal, was introduced for abundant semigroups by El Qallali (see [5]). The construction for abundant semigroups satisfying the following conditions has been given in [5]: S contains a multiplicative type A transversal S; E(S) generates a regular semiband 〈E(S)〉; E(S) is a semilattice transversal of 〈E(S)〉. The purpose of this paper is to give a structure theorem for abundant semigroups with quasi-ideal adequate transversals. Many conditions considered in [5] will be removed. Hence this class of abundant semigroups properly includes the class of abundant semigroups studied in [5] (see [5, Theorem 3.9 and Theorem 4.2]). In Section 1, we collect some basic concepts and results from [3, 4, 5, 6], which are frequently used in this paper. In Section 2, we give some properties about the sets I and Λ, which are two components in the construction for this class of abundant semigroups. In their investigation into the construction for regular semigroups with inverse transversals, McAlister and McFadden proved that I and Λ are R-unipotent and L-unipotent sub-bands respectively of S (see [2, Proposition 1.7]). But the corresponding result in abundant semigroups fails to be true. We give a counterexample to show that I and Λ need not be semigroups if S is an abundant semigroup with a quasiideal adequate transversal. In Section 3, we begin by introducing the concept of (E, I,Λ)-system, then we use it to construct an abundant semigroup containing a quasi-ideal adequate transversal; conversely, every abundant semigroup with a quasiideal adequate transversal can be constructed in this way. Finally, we apply this theorem to a special case. We use the notation and terminology of [8], and assume some familiarity with the contents of [6] and [7].