Inverse semigroups and extensions of groups by semilattices
This is the first of three papers on inverse semigroups published by the authors in J. Algebra [see the following reviews]. Here they generalize the theory of P-semigroups (which characterize E-unitary inverse semigroups) to the non-regular case. The main theorem, from which the McAlister P-Theorem is recovered, is as follows: Let M be a monoid whose idempotents commute. The following are equivalent: (1) There is a surjective morphism \(\phi\) : \(M\to G\) onto a group such that \(1\phi^{-1}=E(M)\). (2) M is E-unitary and dense (meaning that for any \(s\in M\) there exists \(s_ 1,s_ 2\) such that \(ss_ 1,s_ 2s\in E(M))\). (3) M is isomorphic to a monoid C/G where G is a group acting transitively without fixpoints on a connected, idempotent and commutative category C. The group G can be taken as the ``free group over M'' and C the corresponding ``derived covering'': all these terms are defined and developed within the paper.
scientific article; zbMATH DE number 4015008
scientific article