Processing math: 100%

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

Semigroups

\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}


QR Code
QR Code regular_semigroups (generated for current page)