b in Q and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a♯ denotes the group inverse of a in some (hence any) subgroup of Q. If S is a straight left order in Q, then Q is necessarily regular; the idea is that Q has a better understood structure than that of S. Necessary and sufficient conditions exist on a semigroup S for S to be a straight left order. The technique is to consider a pair
We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford’s seminal work describing bisimple inverse monoids
in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford’s
work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work of Gantos
into a wider and clearer context, and pave the way for further progress.
Authors:John Fountain, Gracinda Gomes and Victoria Gould
We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion
for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark
that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of
techniques that may be of interest in their own right.