Semirings with zero

 
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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}}
 \pagestyle{myheadings}\thispagestyle{myheadings}
 \markboth{\today}{math.chapman.edu/structures}
 
 \begin{document}
 \textbf{\Large Semirings with zero}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Semirings_with_zero}{edit}
 
 \abbreviation{SRng$_0$}
 
 \begin{definition}
 A \emph{semiring with zero} is a structure $\mathbf{S}=\langle S,+,0,\cdot
 \rangle $ of type $\left\langle 2,0,2\right\rangle $ such that
 
 
 $\langle S,+,0\rangle $ is a \href{Commutative_monoids.pdf}{commutative monoids}
 
 
 $\langle S,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups}
 
 
 $0$ is a zero for $\cdot$:  $0\cdot x=0$, $x\cdot 0=0$
 
 
 $\cdot$ distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$
 to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 
 
 $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
 \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\\\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.pdf}{Idempotent semirings with zero} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Semirings.pdf}{Semirings} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %

 
 \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 Semirings with zero}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Semirings_with_zero}{edit}
 
 \abbreviation{SRng$_0$}
 
 \begin{definition}
 A \emph{semiring with zero} is a structure $\mathbf{S}=\langle S,+,0,\cdot
 \rangle $ of type $\left\langle 2,0,2\right\rangle $ such that
 
 
 $\langle S,+,0\rangle $ is a \href{Commutative_monoids.pdf}{commutative monoids}
 
 
 $\langle S,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups}
 
 
 $0$ is a zero for $\cdot$:  $0\cdot x=0$, $x\cdot 0=0$
 
 
 $\cdot$ distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$
 to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 
 
 $h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
 \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\\\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.pdf}{Idempotent semirings with zero} 
 
 \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/Semirings_with_zero.pdf

%%run pdflatex

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\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 Semirings with zero}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Semirings_with_zero}{edit}

\abbreviation{SRng$_0$}

\begin{definition}
A \emph{semiring with zero} is a structure $\mathbf{S}=\langle S,+,0,\cdot
\rangle $ of type $\left\langle 2,0,2\right\rangle $ such that


$\langle S,+,0\rangle $ is a \href{Commutative_monoids.pdf}{commutative monoids}


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


$0$ is a zero for $\cdot$:  $0\cdot x=0$, $x\cdot 0=0$


$\cdot$ distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$
\end{definition}

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

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
\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\\\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.pdf}{Idempotent semirings with zero} 

\end{subclasses}
\begin{superclasses}\ 

\href{Semirings.pdf}{Semirings} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%