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A common framework for restriction semigroups and regular ∗semigroups
 J. Pure Applied Algebra
"... a b s t r a c t 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 u ..."
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a b s t r a c t 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 twosided 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 −1 a, or in the twosided version, to both). In this paper the notion of restriction semigroup is generalized to Prestriction semigroup, derived instead from reducts of regular * semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] PEhresmann 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 twosided case, is that for every Prestriction semigroup S there is a Pseparating 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 socalled '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. © 2011 Elsevier B.V. All rights reserved. The various strands in the historical development of the class of restriction semigroups are comprehensively reviewed in In
Semiabundant Semigroups with QuasiEhresmann Transversals
"... Abstract. Chen (Communications in Algebra 27(2), 42754288, 1999) introduced and investigated orthodox transversals of regular semigroups. In this paper, we initiate the investigation of quasiEhresmann transversals of semiabundant semigroups which are generalizations of orthodox transversals of r ..."
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Abstract. Chen (Communications in Algebra 27(2), 42754288, 1999) introduced and investigated orthodox transversals of regular semigroups. In this paper, we initiate the investigation of quasiEhresmann transversals of semiabundant semigroups which are generalizations of orthodox transversals of regular semigroups. Some interesting properties associated with quasiEhresmann transversals are established. Moreover, a structure theorem of semiabundant semigroups with generalized biideal strong quasiEhresmann transversals is obtained. Our results generalize and enrich Chen’s results. 1.