Abbreviation: InvSgrp

An inverse semigroup is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle $ such that

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

$^{-1}$ is an inverse: $xx^{-1}x=x$, $(x^{-1})^{-1}=x$

idempotents commute: $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$

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

$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$

Example 1: $\langle I_X,\circ,^{-1}\rangle$, the symmetric inverse semigroup of all one-to-one partial functions on a set $X$, with composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &52\\ f(5)= &208\\ f(6)= &911\\ f(7)= &\\ \end{array}$

Groups

Commutative inverse semigroups

Semigroups