History of Idempotent semirings with zero

 
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 \begin{document}
 \textbf{\Large Idempotent semirings with zero}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Idempotent_semirings_with_zero}{edit}
 
 \abbreviation{ISRng$_0$}
 
 \begin{definition}
 An \emph{idempotent semiring with zero} is a \href{Semirings_with_zero.pdf}{semirings with zero} $\mathbf{S}=\langle S,\vee,0,\cdot
 \rangle $ such that 
 $\vee$ is idempotent:  $x\vee x=x$
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{S}$ and $\mathbf{T}$ be idempotent semirings with zero. A morphism from $\mathbf{S}$
 to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 
 
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$
 \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}
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 \begin{subclasses}\ 
 
 \href{Idempotent_semirings_with_identity_and_zero.pdf}{Idempotent semirings with identity and zero} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Idempotent_semirings.pdf}{Idempotent semirings} 
 
 \href{Semirings_with_zero.pdf}{Semirings with zero} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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