Semigroups of left quotients: existence, straightness and locality

A subsemigroup <f>S</f> of a semigroup <f>Q</f> is a local left order in <f>Q</f> if, for every group <f>H</f>-class <f>H</f> of <f>Q</f>, <f>S∩H</f> is a left order in <f>H</f> in the sense of group theory. That is, every <f>q∈H</f> can be written as <f>a♯b</f> for some <f>a,b∈S∩H</f>, where <f>a♯</f> denotes the group inverse of <f>a</f> in <f>H</f>. On the other hand, <f>S</f> is a left order in <f>Q</f> and <f>Q</f> is a semigroup of left quotients of <f>S</f> if every element of <f>Q</f> can be written as <f>c♯d</f> where <f>c,d∈S</f> and if, in addition, every element of <f>S</f> that is square cancellable lies in a subgroup of <f>Q</f>. If one also insists that <f>c</f> and <f>d</f> can be chosen such that <f>c R d</f> in <f>Q</f>, then <f>S</f> is said to be a straight left order in <f>Q</f>.This paper investigates the close relation between local left orders and straight left orders in a semigroup <f>Q</f> and gives some quite general conditions for a left order <f>S</f> in <f>Q</f> 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. [Copyright &y& Elsevier]