Mathematical Structures: Semirings with zero

# Semirings with zero

http://mathcs.chapman.edu/structuresold/files/Semirings_with_zero.pdf
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\begin{document}
\textbf{\Large Semirings with zero}

\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/}
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\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}
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