Semigroups of left quotients: existence, straightness and locality

A subsemigroup S of a semigroup Q is a local left order in Q if, for every maximal subgroup H of Q, the intersection of S with H is a local left order in the sense of group theory. That is, every q in H can be written as a#b for some a,b in the intersection of S with H, where here a# denotes the group inverse of a in H. On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c#d where c,d are in S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. If one also insists that c and d can be chosen to be related by Green's relation R in Q, then S is said to be a straight left order in Q. This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper.