# Algebra Seminar talk

2007-05-18

Melanija **Mitrovic***Studying Semigroups through their Semilattice Decomposition*

Abstract:

Following Donald B. McAlister, semigroups are, by definition,
simple objects but the very simplicity of
their definition means that they can have a very complicated and
intricate structure. Various approaches have been developed over the
years to construct frameworks for understanding this structure.
Four methods with general applications are ideal extensions,
semilattice decompositions, subdirect decompositions, and group
coextensions.The aim of this talk is to show how semilattice
decompositions can be used for certain situation.

An element *a* of a semigroup *S* is said to be *regular*
in *S* if there exists *x∈S* such that *a=axa*. Let
Reg(*S*) denote the set of all regular elements of *S*, and for a
subsemigroup *T ⊆ S* let reg(*T*) denote the intersection

*T ∩ Reg(*, that is, the set of all elements of

*S*)*T*which are

regular in *S*. Clearly,

Reg(*T*) ⊆ reg(*T*)

and the inclusion can be strict (since every semigroup can be
embedded into a regular one). However, for certain kinds of
subsemigroups *T*, the inclusion above becomes an equality: for
instance, this is the case if *T* is a two-sided ideal or a local
submonoid of *S*. A natural question in this directions is the
following: what are semigroups *S* such that Reg(*T*)=reg(*T*) for
every subsemigroup *T* of *S*. Its solution gives an interesting
bridge between semilattice decompositions of semigroups on the one
side and hereditary properties of semigroups on the other.