−Table of Contents
Regular semigroups
Abbreviation: RSgrp
Definition
An element x of a semigroup S is said to be \emph{regular} if exists y in S such that xyx=x.
Definition
A \emph{regular semigroup} is a semigroups S=⟨S,⋅⟩ such that each element is regular.
Definition
A \emph{regular semigroup} is a structure S=⟨S,⋅⟩, where ⋅ is an infix binary operation, called the \emph{semigroup product}, such that
⋅ is associative: (xy)z=x(yz)
each element is \emph{regular}: ∃y(xyx=x)
Definition
We say that y is an \emph{inverse} of an element x in a semigroup S if x=xyx and y=yxy.
Morphisms
Let S and T be regular semigroups. A morphism from S to T is a function h:SarrowT that is a homomorphism:
h(xy)=h(x)h(y)
Examples
Example 1: ⟨TX,∘⟩, the \emph{full transformation semigroup} of functions on X, with composition.
⟨End(V),∘⟩, the \emph{endomorphism monoid} of a vector space V, with composition.
Basic results
If x is a regular element of a semigroup (say x=xyx), then x has an inverse, namely yxy, since x=x(yxy)x and yxy=(yxy)x(yxy).
Properties
Finite members
f(1)=1f(2)=3f(3)=9f(4)=42f(5)=206f(6)=1352f(7)=10168f(8)=91073f(9)=925044
(the opposite of a semigroup S is identified with S in the table above, see https://oeis.org/A001427)
Subclasses
Superclasses
\begin{bibdiv} \begin{biblist}
\bib{MR1455373}{book}{
author={Howie, John M.}, title={Fundamentals of semigroup theory}, series={London Mathematical Society Monographs. New Series}, volume={12}, note={Oxford Science Publications}, publisher={The Clarendon Press Oxford University Press}, place={New York}, date={1995}, pages={x+351}, isbn={0-19-851194-9}, review={\MR{1455373 (98e:20059)}},
}
\end{biblist} \end{bibdiv}