Restriction Semigroups: Structure, Varieties and Presentations

Classes of (left) restriction semigroups arise from partial transformation monoids and form a wider class than inverse semigroups. Firstly, we produce a presentation of the Szendrei expansion of a monoid, which is a left restriction monoid, using a similar approach to Exel's presentation for the Szendrei expansion of a group. Presentations for the Szendrei expansion of an arbitrary left restriction semigroup and of an inverse semigroup are also found. For our second set of results we look at structure theorems, or P-theorems, for proper restriction semigroups and produce results in a number of ways. Initially, we generalise Lawson's approach for the proper ample case, in which he adapted the one-sided result for proper left ample semigroups. The awkwardness of this approach illustrates the need for a symmetrical two-sided result. Creating a construction from partial actions, based on the idea of a double action, we produce structure theorems for proper restriction semigroups. We also consider another construction based on double actions which yields a structure theorem for a particular class of restriction semigroups. In fact, this was our first idea, but the class of proper restriction semigroups it produces is not the whole class. For our final topic we consider varieties of left restriction semigroups. Specifically, we shall show that the class of (left) restriction semigroups having a cover over a variety of monoids is a variety of (left) restriction semigroups. We do this in two ways. Generalising results by Gomes and Gould on graph expansions, we consider the graph expansion of a monoid and obtain our result for the class of left restriction monoids. Following the same approach as Petrich and Reilly we produce the result for the class of left restriction semigroups and for the class of restriction semigroups.