Orders and straight left orders in completely regular semigroups (Q1596338)

A subsemigroup \(S\) of a completely regular semigroup \(Q\) is a left order in \(Q\) if every element of \(Q\) can be written as \(a^\#b\) where \(a,b\in S\) and \(a^\#\) is the inverse of \(a\) in a subgroup of \(Q\). If \(a,b\) can always be chosen so that \(a{\mathcal R}b\) in \(Q\), then \(S\) is called a straight left order in \(Q\). The main result of this paper characterizes those semigroups that are straight left orders in completely regular semigroups: they are semilattices of matrices of right reversible cancellative semigroups satisfying some further conditions. It is also proved that every two-sided order in a completely regular semigroup is straight. These two results yield then a characterisation of two-sided orders in completely regular semigroups.