### 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.