Mathematical Structures: Order algebras

# Order algebras

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

\abbreviation{OrdA}
\begin{definition}
An \emph{order algebra} is a structure $\mathbf{A}=\langle A,\cdot \rangle$, where $\cdot$ is an infix binary operation such that

$\cdot$ is idempotent:  $x\cdot x=x$

$(x\cdot y)\cdot x=y\cdot x$

$(x\cdot y)\cdot y=x\cdot y$

$x\cdot ((x\cdot y)\cdot z)=x\cdot(y\cdot z)$

$((x\cdot y)\cdot z)\cdot y=(x\cdot z)\cdot y$

Remark:

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be order algebras. A morphism from $\mathbf{A}$
to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

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

\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 & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & \\\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 & \\\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 $n$.

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

\href{Bands.pdf}{Bands}

\end{subclasses}
\begin{superclasses}\

\href{Groupoids.pdf}{Groupoids}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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