Semigroups of straight left inverse quotients

Abstract

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as a^{-1}b, where a, b in S and a^{-1} is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be mathscr {R}-related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of {mathscr {R}} and {mathscr {L}} from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the mathscr {L}-classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight.