Algebra Seminar talk
Studying Semigroups through their Semilattice Decomposition
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 intersectionT ∩ Reg(S), that is, the set of all elements of 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.