A Common Framework for Restriction Semigroups and Regular *-Semigroups

Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sided versions of the restriction property. From a varietal perspective, these classes of semigroups – more generally, the corresponding classes of Ehresmann semigroups – derive from reducts of inverse semigroups, now taking a to a = aa−1 (or, dually, to a∗ = a−1a, or in the two-sided version, to both). In this paper the notion of restriction semigroup is generalized to P -restriction semigroup, derived instead from reducts of regular ∗-semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] P -Ehresmann semigroups. The first main theorem is an abstract characterization of the posets P of projections of each type of such semigroup as ‘projection algebras’. The second main theorem, at least in the two-sided case, is that for every P -restriction semigroup S there is a P -separating representation into a regular ∗-semigroup, namely the ‘Munn’ semigroup on its projection algebra, consisting of the isomorphisms between the algebra’s principal ideals under a modified composition. This theorem specializes to known results for restriction semigroups and for regular ∗-semigroups. A consequence of this representation is that projection algebras also characterize the posets of projections of regular ∗-semigroups. By further characterizing the sets of projections ‘internally’, we connect our universal algebraic approach with the classical approach of the so-called ‘York school’. The representation theorem will be used in a sequel to show how the structure of the free members in some natural varieties of (P -) restriction semigroups may easily be deduced from the known structure of associated free inverse semigroups. The various strands in the historical development of the class of restriction semigroups are comprehensively reviewed in [9], [10] and [7] (see later in this introduction) but the inspiration for the current work comes in particular from [9]. As noted above, the left restriction semigroups model unary semigroups of partial mappings on a set, with α+ the identity map on the domain of α. The set of ‘distinguished’ idempotents that results need not comprise all idempotents of the semigroup. In [9], V. Gould formalized the connection with the so-called ‘York school’: the left restriction semigroups are the weakly left E-ample semigroups S, defined in terms