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 group H-class H of Q, S ∩ H is a left order in H in the sense of group theory. That is, every q ∈ H can be written as ab for some a, b ∈ S ∩ H , where 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 cd where c, d ∈ 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 such that cR d 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 in Q 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.