Mathematical Structures: Commutative lattice-ordered semigroups

# Commutative lattice-ordered semigroups

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Changed: 28,29c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit}
 \Large Commutative lattice-ordered semigroups \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_lattice-ordered_semigroups}{edit}

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

Changed: 34,37c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 A commutative lattice-ordered semigroup is a \href{lattice-ordered_semigroups.PDF}{lattice-ordered semigroup} $\mathbf{A}=\langle A,\vee,\wedge,\cdot\rangle$ such that

Changed: 39,41c36
 $op_1$ is (name of property): $axiom_1$ $op_2$ is ...: $...$
 $\cdot$ is commutative: $xy=yx$

Changed: 50,51c45,48
 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 commutative lattice-ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$.

Changed: 77c74
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & variety \\\hline

Changed: 83,84c80,81
 Congruence distributive & \\\hline Congruence modular & \\\hline
 Congruence distributive & yes \\\hline Congruence modular & yes \\\hline

Changed: 119,121c116
 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Commutative_residuated_lattices.pdf}{Commutative residuated lattices} expansion

Changed: 127c122,124
 \href{....pdf}{...} supervariety
 \href{Lattice-ordered_semigroups.pdf}{Lattice-ordered semigroups} supervariety \href{Commutative_semigroups.pdf}{Commutative semigroups} subreduct

Changed: 129c126
 \href{....pdf}{...} subreduct
 \href{Lattices.pdf}{Lattices} subreduct

http://mathcs.chapman.edu/structuresold/files/Commutative_lattice-ordered_semigroups.pdf
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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\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/}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Commutative lattice-ordered semigroups}

\abbreviation{CLSgrp}

\begin{definition}
A \emph{commutative lattice-ordered semigroup} is a \href{lattice-ordered_semigroups.PDF}{lattice-ordered semigroup} $\mathbf{A}=\langle A,\vee,\wedge,\cdot\rangle$ such that

$\cdot$ is \emph{commutative}:  $xy=yx$

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 commutative lattice-ordered semigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x \vee y)=h(x) \vee h(y)$,
$h(x \wedge y)=h(x) \wedge h(y)$,
$h(x \cdot y)=h(x) \cdot h(y)$.
\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                       & variety \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & yes \\\hline
Congruence modular              & yes \\\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_residuated_lattices.pdf}{Commutative residuated lattices} expansion

\end{subclasses}

\begin{superclasses}\

\href{Lattice-ordered_semigroups.pdf}{Lattice-ordered semigroups} supervariety

\href{Commutative_semigroups.pdf}{Commutative semigroups} subreduct

\href{Lattices.pdf}{Lattices} subreduct

\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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