Orders and Straight Left Orders in Completely Regular Semigroups

Abstract

A subsemigroup S of a completely regular semigroup Q is an order in Q if every element of Q can be written as a 5 b and as cd 5 where aY bY cY d P S and x 5 is the inverse of x P Q in a subgroup of Q. If only the ®rst condition holds and one insists also that a R b in Q, then S is said to be a straight left order in Q. This paper characterizes those semigroups that are straight left orders in completely regular semigroups. A consequence of this result, together with some technicalities concerning lifting of morphisms, is a description of orders in completely regular semigroups.2000 Mathematics Subject Classi®cation: 20 M 10

Semigroups of Left Quotients—The Layered Approach

“…Theorem 5.5 of [15] (see also [17]) is the specialisation of Theorem 6.1 to the case where each S α is a straight left order in a completely simple semigroup T α . Surprisingly, the conditions in (A) are only slightly stronger than those in Theorem 5.5 of [15]. In the latter result one can use the decomposition of each S α as a matrix of right reversible cancellative subsemigroups to deduce that (≤ ℓ , ≤ r ) is a * -pair from the assumptions that ≤ ℓ and ≤ r are transistive, R ′ is a left congruence, L ′ ⊆ L * and R ′ ⊆ R * [15].…”

“…We remark here that if S is a left order in Q, H is a congruence on Q and Q is regular, then S is perforce straight [10]. Subsequently, a number of papers have appeared characterising semigroups that are (straight) left orders in semigroups in various classes, for example [12,15]. More recently, Theorem 4.4 of [14] gives a characterisation of straight left orders in arbitrary semigroups; we recap briefly below the approach of [14].…”

“…The latter are known by the theorem of Ore and Dubreil to be left orders in groups [2]. A similar approach is taken to straight left orders in completely simple semigroups in [15]. The approach of [12,15] and other early papers was thus to determine the global structure of a semigroup from the local structure.…”

“…A similar approach is taken to straight left orders in completely simple semigroups in [15]. The approach of [12,15] and other early papers was thus to determine the global structure of a semigroup from the local structure. Other results, for example Theorem 5.12 of [8], do not use this approach explicitly, but nevertheless, the semigroups in question have a relative ideal series for which the Rees quotients are left orders in completely 0-simple semigroups.…”

“…To this end we introduce the idea of a partial order decomposition of a semigroup, that allows us to consider at the same time semilattice decompositions and decompositions into Rees quotients. This technique enables us to extend the main theorem of [15] to arbitrary semilattices of orders, find an alternative proof of Theorem 5.12 of [8], and a new characterisation of straight left orders in completely semisimple semigroups.…”