Structure theorems for proper restriction semigroups (Claire Cornock, NBSAN York, 25th November 2009)

Abstract: Following on from McAlister's P-theorem for proper inverse semigroups, we consider P-theorems for proper restriction semigroups. One-sided restriction semigroups arise in the work of a number of authors; structure theorems exist for proper left restriction semigroups (and special subclasses) analogous to those of McAlister for proper inverse semigroups.

We use the one sided results to produce two sided results that are left-right symmetric, motivated by the idea of a double action of a monoid on a semilattice. Imposing further conditions on the monoid we then obtain symmetrical structure theorems for proper weakly ample and proper ample semigroups.