Idempotent semirings

 
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 \newtheorem{definition}{Definition}
 \newtheorem*{morphisms}{Morphisms}
 \newtheorem*{basic_results}{Basic Results}
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 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
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 \begin{document}
 \textbf{\Large Idempotent semirings}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Idempotent_semirings}{edit}
 
 \abbreviation{ISRng}
 \begin{definition}
 An \emph{idempotent semiring} is a semiring $\mathbf{S}=\left\langle S,\vee
 ,\cdot \right\rangle $ such that
 
 
 $\vee $ is idempotent:  $x\vee x=x$
 
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{S}$ and $\mathbf{T}$ be idempotent semirings. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
 homomorphism: 
 
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
 
 \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 & Decidable\\\hline
 Quasiequational theory & \\\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 & \\\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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &1\\
 f(3)= &\\
 f(4)= &\\
 f(5)= &\\
 f(6)= &\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Idempotent_semirings_with_zero_and_one.pdf}{Idempotent semirings with zero and one} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Semirings.pdf}{Semirings} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

http://mathcs.chapman.edu/structuresold/files/Idempotent_semirings.pdf

%%run pdflatex

%

\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
\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}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Idempotent semirings}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Idempotent_semirings}{edit}

\abbreviation{ISRng}
\begin{definition}
An \emph{idempotent semiring} is a semiring $\mathbf{S}=\left\langle S,\vee
,\cdot \right\rangle $ such that


$\vee $ is idempotent:  $x\vee x=x$

\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be idempotent semirings. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

\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 & Decidable\\\hline
Quasiequational theory & \\\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 & \\\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 $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Idempotent_semirings_with_zero_and_one.pdf}{Idempotent semirings with zero and one} 

\end{subclasses}
\begin{superclasses}\ 

\href{Semirings.pdf}{Semirings} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%