Mathematical Structures: Idempotent semirings with identity

# Idempotent semirings with identity

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http://mathcs.chapman.edu/structuresold/files/Idempotent_semirings_with_identity.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Idempotent semirings with identity}

\abbreviation{ISRng$_1$}

\begin{definition}
An \emph{idempotent semiring with identity} is a \href{Semirings_with_identity.pdf}{semirings with identity} $\mathbf{S}=\langle S,\vee,\cdot,1 \rangle$ such that

$\vee$ is idempotent:  $x\vee x=x$
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
\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/}
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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_identity.pdf}{Semirings with identity}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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Edited March 25, 2003 11:27 am by Peter Jipsen (diff)
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