Identities in the Algebra of Partial Maps

We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable

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

Partial maps with domain and range: extending Schein's representation

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised