Abbreviation: RSgrp

An element $x$ of a semigroup $S$ is said to be \emph{regular} if exists $y$ in $S$ such that $xyx=x$.

A \emph{regular semigroup} is a semigroups $\mathbf{S}=\langle S,\cdot \rangle $ such that each element is regular.

A \emph{regular semigroup} is a structure $\mathbf{S}=\langle S,\cdot \rangle $, where $\cdot $ is an infix binary operation, called the \emph{semigroup product}, such that

$\cdot $ is associative: $(xy)z=x(yz)$

each element is \emph{regular}: $\exists y(xyx=x)$

We say that $y$ is an \emph{inverse} of an element $x$ in a semigroup $S$ if $x=xyx$ and $y=yxy$.

Let $\mathbf{S}$ and $\mathbf{T}$ be regular semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$

Example 1: $\langle T_X,\circ\rangle $, the \emph{full transformation semigroup} of functions on $X$, with composition.

$\langle End(V),\circ\rangle $, the \emph{endomorphism monoid} of a vector space $V$, with composition.

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 $yxy=(yxy)x(yxy)$.

$\begin{array}{lr} f(1)= &1
f(2)= &3
f(3)= &9
f(4)= &42
f(5)= &206
f(6)= &1352
f(7)= &10168
f(8)= &91073
f(9)= &925044 \end{array}$

(the opposite of a semigroup $S$ is identified with $S$ in the table above, see https://oeis.org/A001427)

Bands

Inverse semigroups

Completely regular semigroups

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}