Abbreviation: CInvSgrp

A \emph{commutative inverse semigroup} is an inverse semigroups $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that

$\cdot$ is commutative: $xy=yx$

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

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

$\cdot$ is commutative: $xy=yx$

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

Let $\mathbf{S}$ and $\mathbf{T}$ be commutative 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:

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

Abelian groups

Semilattices

Inverse semigroups