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 ^# b and as cd ^# where and x ^# is the inverse of in a subgroup of Q. If only the first condition holds and one insists also that a?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.