Orders and straight left orders in completely regular semigroups

A subsemigroup S of a completely regular semigroup Q is an order in Q if ev ery element of Q can be written as a(#)b and as cd(#) where a, b, c,d is an element of S and x(#) is the inverse of x is an element of Q in ri subgrou p of e. If only the first condition holds and one insists also that a R b i n e, then S is said to be a straight left order in e. This paper characteri zes those semigroups that are straight left orders in completely regular se migroups. A consequence of this result, together with some technicalities c oncerning lifting of morphisms, is a description of orders in completely re gular semigroups. 2000 Mathematics Subject Classification: 20 M 10.