Mathematical Structures: Inverse semigroups

# Inverse semigroups

http://mathcs.chapman.edu/structuresold/files/Inverse_semigroups.pdf
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\begin{document}
\textbf{\Large Inverse semigroups}

\abbreviation{InvSgrp}
\begin{definition}
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:  $xx^{-1}x=x$, $(x^{-1})^{-1}=x$

idempotents commute:  $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$
\end{definition}
\begin{morphisms}
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:

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

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\langle I_X,\circ,^{-1}\rangle$, 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.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & Variety\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & No\\\hline
Residual size & \\\hline
Congruence distributive & No\\\hline
Congruence modular & No\\\hline
Congruence n-permutable & No\\\hline
Congruence regular & No\\\hline
Congruence uniform & No\\\hline
Congruence extension property & No\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & No\\\hline
Amalgamation property & Yes\\\hline
Strong amalgamation property & Yes\\\hline
Epimorphisms are surjective & Yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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

\href{Groups.pdf}{Groups}

\href{Commutative_inverse_semigroups.pdf}{Commutative inverse semigroups}

\end{subclasses}
\begin{superclasses}\

\href{Semigroups.pdf}{Semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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