Mathematical Structures: Partially ordered semigroups

# Partially ordered semigroups

Difference (from prior major revision) (no other diffs)

Changed: 28,29c28,29

Changed: 31c31
 \abbreviation{Abbr}
 \abbreviation{PoSgrp}

Changed: 34,35c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that
 A partially ordered semigroup is a structure $\mathbf{A}=\langle A,\cdot,\le\rangle$ such that

Changed: 37c36
 $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 $\langle A,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroup}

Changed: 39c38
 $op_1$ is (name of property): $axiom_1$
 $\langle G,\le\rangle$ is a \href{Partially_ordered_sets.pdf}{partially ordered set}

Changed: 41c40
 $op_2$ is ...: $...$
 $\cdot$ is orderpreserving: $x\le y\implies xz\le yz \text{ and } zx\le zy$

Changed: 50,51c49,51
 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 Let $\mathbf{A}$ and $\mathbf{B}$ be partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $x\le y\implies h(x)\le h(y)$

Changed: 77c77
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & quasivariety \\\hline

Changed: 119c119
 \href{....pdf}{...} subvariety
 \href{Commutative_partially_ordered_semigroups.pdf}{Commutative partially ordered semigroups}

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 \href{....pdf}{...} expansion
 \href{Lattice-ordered_semigroups.pdf}{Lattice-ordered semigroups} expanded type

Changed: 127,129c127
 \href{....pdf}{...} supervariety \href{....pdf}{...} subreduct
 \href{Partially_ordered_groupoids.pdf}{Partially ordered groupoids}

http://mathcs.chapman.edu/structuresold/files/Partially_ordered_semigroups.pdf
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\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{document}
\textbf{\Large Partially ordered semigroups}

\abbreviation{PoSgrp}

\begin{definition}
A \emph{partially ordered semigroup} is a structure $\mathbf{A}=\langle A,\cdot,\le\rangle$ such that

$\langle A,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroup}

$\langle G,\le\rangle$ is a \href{Partially_ordered_sets.pdf}{partially ordered set}

$\cdot$ is \emph{orderpreserving}:  $x\le y\implies xz\le yz \text{ and } zx\le zy$

Remark: This is a template.
If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism:
$h(x \cdot y)=h(x) \cdot h(y)$,
$x\le y\implies h(x)\le h(y)$
\end{morphisms}

\begin{definition}
A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\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})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & quasivariety \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & \\\hline
Congruence $n$-permutable       & \\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\end{finite_members}

\begin{subclasses}\

\href{Commutative_partially_ordered_semigroups.pdf}{Commutative partially ordered semigroups}

\href{Lattice-ordered_semigroups.pdf}{Lattice-ordered semigroups} expanded type

\end{subclasses}

\begin{superclasses}\

\href{Partially_ordered_groupoids.pdf}{Partially ordered groupoids}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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