http://www.macs.hw.ac.uk/~markl/Lisbon2010.pdf

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister's theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

/pdf/enlargements-of-regular-semigroups-4ppjvia7pn.pdf

Enlargements of regular semigroups

Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants.

Variants of Regular Semigroups

Rees matrix covers for regular semigroups

BYD. B. MCALISTERAbstract. A regular semigroup S is locally inverse if each local submonoid eSe, ean idempotent, is an inverse semigroup. It is shown that every locally inversesemigroup is an image of a regular Rees matrix semigroup, over an inversesemigroup, by a homomorphism 0 which is one-to-one on each local submonoid;such a homomorphism is called a local isomorphism. Regular semigroups which arelocally isomorphic images of regular Rees matrix semigroups over semilattices arealso characterized.

/pdf/rees-matrix-covers-for-locally-inverse-semigroups-58uhflax7s.pdf

D. B. McAlister

Papers

Rees matrix covers for locally inverse semigroups