Abbreviation: InvSgrp
An \emph{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: $xx^{-1}x=x$ and $(x^{-1})^{-1}=x$
idempotents commute: $xx^{-1}yy^{-1}=yy^{-1}xx^{-1}$
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 \emph{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.
$x*x=x \implies \exists y\ x=y*y^{-1}$
$\forall x\exists y\ xx^{-1}=y^{-1}y$
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &5
f(4)= &16
f(5)= &52
f(6)= &208
f(7)= &911
f(8)= &4637
f(9)= &26422
f(10)= &169163
f(11)= &1198651
f(12)= &9324047
f(13)= &78860687
f(14)= &719606005
f(15)= &7035514642
\end{array}$