Mathematical Structures: Semigroups with zero

Semigroups with zero

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

\abbreviation{Sgrp$_0$}
\begin{definition}
A \emph{semigroup with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle$ such that

$\langle S,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups}

$0$ is a zero for $\cdot$:  $x\cdot 0=0$, $0\cdot x=0$
\end{definition}

\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be semigroups with zero. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
\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 PTIME\\\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. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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

\href{Commutative_semigroups_with_zero.pdf}{Commutative semigroups with zero}

\end{subclasses}
\begin{superclasses}\

\href{Semigroups.pdf}{Semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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