Inverse semigroups with zero: covers and their structure

Abstract

We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroup is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all £*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical £*-unitary semigroups recently considered by Gomes and Howie.1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18.

Tiling Semigroups

“…In summary: if ⌺ has cardinality n, then w x in Theorem 9.5.6 of 9 it is proved that the tiling semigroup S S is a Rees quotient of a semigroup K , called a Kachel semigroup on n letters. 1 The n semigroup K may be embedded in the McAlister semigroup M , intron n w x duced in 8 .…”

Graph Products of Right Cancellative Monoids

“…In general, prehomomorphisms between inverse monoids are defined in terms of the natural order on the monoids, but the general definition is equivalent to condition (3) when the codomain is a group with zero adjoined. Implicit in [3] is the result that an inverse semigroup with zero is strongly E * -unitary if and only if it is a Rees quotient of an E-unitary inverse semigroup. This was made explicit with an easy proof in [29].…”

Semilattice Ordered Inverse Semigroups

“…Since there are lots of finite E-unitary inverse semigroups and every Rees factor of such a semigroup is E * -unitary, [2], it follows that finite ∨-semilatticed inverse semigroups are very common. [Bulman-Flemming, Fountain, and Gould [2] have also shown that not every E * -unitary inverse semigroup is a Rees factor of an E-unitary inverse semigroup.]…”

“…[Bulman-Flemming, Fountain, and Gould [2] have also shown that not every E * -unitary inverse semigroup is a Rees factor of an E-unitary inverse semigroup.] A class of interesting examples of such semigroups is the following.…”