Mathematical Structures: Goedel algebras

Goedel algebras

http://mathcs.chapman.edu/structuresold/files/Goedel_algebras.pdf
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\begin{document}
\textbf{\Large G\"odel algebras}

\abbreviation{G\"odA}

\begin{definition}
A \emph{G\"odel algebra} is a \href{Heyting_algebras.pdf}{Heyting algebras} $\mathbf{A}=\langle A,\vee,0,\wedge,1,\rightarrow\rangle$ such that

$(x\to y)\vee(y\to x)=1$

Remark:
G\"odel algebras are also called \emph{linear Heyting algebras} since subdirectly irreducible G\"odel algebras are linearly ordered Heyting algebras.
\end{definition}

\begin{definition}
A \emph{G\"odel algebra} is a \href{Representable_FLew-algebras.pdf}{representable FLew-algebra} $\mathbf{A}=\langle A, \vee, 0, \wedge, 1, \cdot, \rightarrow\rangle$ such that

$x\wedge y=x\cdot y$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be G\"odel algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(0)=0$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
\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 & decidable\\\hline
First-order theory & \\\hline
Locally finite & \\\hline
Residual size & countable\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence e-regular & yes, $e=1$\\\hline
Congruence uniform & \\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & yes\\\hline
Equationally def. pr. cong. & yes\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Boolean_algebras.pdf}{Boolean algebras}

\end{subclasses}
\begin{superclasses}\

\href{Heyting_algebras.pdf}{Heyting algebras}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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