Mathematical Structures: Semigroups

# Semigroups

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

\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}

\end{thebibliography}

\end{document}
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