Semigroups

http://mathcs.chapman.edu/structuresold/files/Semigroups.pdf

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

\abbreviation{Sgrp}
\begin{definition}
A \emph{semigroup} is a structure $\mathbf{S}=\left\langle S,\cdot
\right\rangle $, where $\cdot $ is an infix binary operation, called the 
\emph{semigroup product}, such that


$\cdot $ is associative:  $(xy)z=x(yz)$.
\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be semigroups. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 

$h(xy)=h(x)h(y)$

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\left\langle X^{X},\circ \right\rangle $, the collection of functions on a
sets $X$, with composition.

\end{example}
\begin{example}
$\left\langle \Sigma ^{+},\cdot \right\rangle $, the collection of nonempty
strings over $\Sigma $, with concatenation.

\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 & decidable in polynomial time\\\hline
Quasiequational theory & undecidable\\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\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 & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & no\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & no\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &5\\
f(3)= &24\\
f(4)= &188\\
f(5)= &1915\\
f(6)= &28634\\
f(7)= &1627672\\
\end{array}$

\href{Search_for_finite_semigroups.pdf}{Search for finite semigroups}
[http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027851 Semigroups in the Encyclopedia of Integer Sequences]
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\ 

\href{Bands.pdf}{Bands} 

\href{Commutative_semigroups.pdf}{Commutative semigroups} 

\href{Monoids.pdf}{Monoids} 

\href{Semigroups_with_zero.pdf}{Semigroups with zero} 

\end{subclasses}
\begin{superclasses}\ 

\href{Groupoids.pdf}{Groupoids} 

\href{Partial_semigroups.pdf}{Partial semigroups} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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