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\import{algebra/magma.tex}
\section{Semigroups}
\begin{struct}\label{semigroup}
A semigroup $A$ is a magma such that
\begin{enumerate}
%\item for all $a, b, c\in \carrier[A]$ we have $\mul[A](a,\mul[A](b,c)) = \mul[A](\mul[A](a,b),c)$.
\item\label{semigroup_assoc} for all $a, b, c$ we have $\mul[A](a,\mul[A](b,c)) = \mul[A](\mul[A](a,b),c)$.
\end{enumerate}
\end{struct}
\section{Regular semigroups}
\begin{struct}\label{regularsemigroup}
A regular semigroup $A$ is a semigroup such that
\begin{enumerate}
%\item for all $a\in \carrier[A]$ there exists $b\in\carrier[A]$ such that $\mul[A](a, \mul[A](b, a)) = a$.
\item\label{regularsemigroup_regular} for all $a$ there exists $b\in\carrier[A]$ such that $\mul[A](a, \mul[A](b, a)) = a$.
\end{enumerate}
\end{struct}
\section{Inverse semigroups}
\begin{struct}\label{inversesemigroup}
An inverse semigroup $A$ is a regular semigroup such that
\begin{enumerate}
\item\label{inversesemigroup_comm} for all $a,b\in\idempotents{A}$ we have $\mul[A](a, b) = \mul[A](b, a)$.
\end{enumerate}
\end{struct}
\begin{proposition}\label{inversesemigroup_is_semigroup}
Suppose $A$ is an inverse semigroup.
Then $A$ is a semigroup.
\end{proposition}
\begin{proposition}\label{inversesemigroup_is_regularsemigroup}
Suppose $A$ is an inverse semigroup.
Then $A$ is a regular semigroup.
\end{proposition}
\begin{proposition}\label{idempotentelems_eq_iff_orbits_eq}
Let $A$ be an inverse semigroup.
Let $e,f\in\idempotents{A}$.
% TODO use Orbits explicitly?
Suppose for all $x\in\carrier[A]$ there exists $y\in\carrier[A]$
such that $x\cdot e = y\cdot f$.
Suppose for all $x\in\carrier[A]$ there exists $y\in\carrier[A]$
such that $x\cdot f = y\cdot e$.
Then $e = f$.
\end{proposition}
\begin{proof}
Take $x, y\in\carrier[A]$ such that $e = x\cdot f$ and $f = y\cdot e$ by \cref{idempotents}.
%
\begin{align*}
e
&= x \cdot f
\explanation{by assumption}\\
&= x\cdot (f\cdot f)
\explanation{by \cref{idempotents}}\\
&= (x\cdot f)\cdot f
\explanation{by \cref{semigroup_assoc,inversesemigroup_is_semigroup}}\\
&= e\cdot f
\explanation{by assumption}\\
&= f\cdot e
\explanation{by \hyperref[inversesemigroup_comm]{commutativity of idempotent elements}}\\
&= (y\cdot e)\cdot e
\explanation{by assumption}\\
&= y\cdot (e\cdot e)
\explanation{by \cref{semigroup_assoc,inversesemigroup_is_semigroup}}\\
&= y \cdot e
\explanation{by \cref{idempotents}}\\
&= f
\explanation{by assumption}
\end{align*}
\end{proof}
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