Unit orthodox semigroups

Abstract

Let S be a regular semigroup. Given x ∈ S , we shall say that a ∈ S is an associate of x if xax = x . The set of associates of x ∈ S will be denoted by A ( x ). Now suppose that S has an identity element 1. Let H 1 denote the group of units of S . Then we say that u ∈ S is a unit associate of x whenever u ∈ A ( x )∩ H l . In what follows we shall write U ( x ) = A ( x )∩= H 1 , and we shall say that S is unit regular [1, 3] if (∀ x ∈ S ) U ( x )≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x ∈ S , the set U ( x ) is a singleton). It is worthy of mention that neither of the examples cited above is of this type.