History of Inverse semigroups

 
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 \begin{document}
 \textbf{\Large Inverse semigroups}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Inverse_semigroups}{edit}
 
 \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}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
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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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