Mathematical Structures: Function rings

# Function rings

http://mathcs.chapman.edu/structuresold/files/Function_rings.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Function rings}

\abbreviation{FRng}
\begin{definition}
A \emph{function ring} (or $f$\emph{-ring}) is an
\href{Lattice-ordered_rings.pdf}{Lattice-ordered rings} $\mathbf{F}=\langle F,\vee,\wedge,+,-,0,\cdot\rangle$ such that

$x\wedge y=0$, $z\ge 0\ \implies\ x\cdot z\wedge y=0$, $z\cdot x\wedge y=0$

Remark:

\end{definition}
\begin{definition}
\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be $f$-rings. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a
homomorphism: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x+y)=f(x)+f(y)$.
\end{morphisms}
\begin{basic_results}
The variety of $f$-rings is generated by the class of linearly ordered $\ell$-rings.
This means $f$-rings are subdirect products of linearly ordered $\ell$-rings, i.e. $f$-rings are representable $\ell$-rings (see e.g. [G. Birkhoff, Lattice Theory, 1967]).
\end{basic_results}
\begin{examples}
\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 & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Congruence distributive & yes, see \href{Lattices.pdf}{lattices}\\\hline
Congruence extension property & \\\hline
Congruence n-permutable & yes, $n=2$, see \href{Groups.pdf}{groups}\\\hline
Congruence regular & yes, see \href{Groups.pdf}{groups}\\\hline
Congruence uniform & yes, see \href{Groups.pdf}{groups}\\\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 $n$.

$\begin{array}{lr} Only the one-element$f$-ring. \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Commutative_function_rings.pdf}{Commutative function rings}

\end{subclasses}
\begin{superclasses}\

\href{Lattice-ordered_rings.pdf}{Lattice-ordered rings}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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