Mathematical Structures: Heyting algebras

# Heyting algebras

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http://mathcs.chapman.edu/structuresold/files/Heyting_algebras.pdf
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\begin{document}
\textbf{\Large Heyting algebras}

\abbreviation{HA}
\begin{definition}
A \emph{Heyting algebra} is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\rightarrow \right\rangle$ such that

$\left\langle A,\vee ,0,\wedge ,1\right\rangle$ is a bounded distributive
lattice

$\rightarrow$ gives the residual of $\wedge$:  $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$

\end{definition}
\begin{definition}
A \emph{Heyting algebra} is a FLew-algebra $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that

$x\wedge y=x\cdot y$

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Heyting 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 & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\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 & no\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & yes\\\hline
Equationally def. pr. cong. & yes\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &3\\ f(6)= &5\\ f(7)= &8\\ f(8)= &15\\ f(9)= &26\\ f(10)= &47\\ f(11)= &82\\ f(12)= &151\\ f(13)= &269\\ f(14)= &494\\ f(15)= &891\\ f(16)= &1639\\ f(17)= &2978\\ f(18)= &5483\\ f(19)= &10006\\ f(20)= &18428\\ \end{array}$

Values known up to size 49 Marcel Ern\'e;, Jobst Heitzig and J\"urgen Reinhold,\emph{On the number of distributive lattices},
Electron. J. Combin.,
\textbf{9}2002,Research Paper 24, 23 pp. (electronic)\href{http://www.ams.org/mathscinet-getitem?mr=2003c:05012}{MRreview}
\end{finite_members}
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\begin{subclasses}\

\href{Goedel_algebras.pdf}{Goedel algebras}

\end{subclasses}
\begin{superclasses}\

\href{Bounded_distributive_lattices.pdf}{Bounded distributive lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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