Orders and Straight Left Orders in Completely Regular Semigroups

 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and x# is the inverse of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} 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.