Brouwerian semilattices

http://math.chapman.edu/structuresold/files/Brouwerian_semilattices.pdf

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\begin{document}
\textbf{\Large Brouwerian semilattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Brouwerian_semilattices}{edit}

\abbreviation{BrSlat}

\begin{definition}
A \emph{Brouwerian semilattice} is a structure $\mathbf{A}=\langle A, \wedge, 1, \rightarrow\rangle$ such that

$\langle A, \wedge, 1\rangle$ is a \href{Semilattices_with_identity.pdf}{semilattice with identity}

$\rightarrow$ gives the residual of $\wedge$:  $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$
\end{definition}

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

$h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
\end{morphisms}

\begin{definition}
A \emph{Brouwerian semilattice} is a \href{Hoops.pdf}{hoop} $\mathbf{A}=\langle A, \cdot, 1, \rightarrow\rangle$ such that

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

\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 & \\\hline
Locally finite & yes\\\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 & \\\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 $n$.

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

Values known up to size 49 \cite{ErneHeitzigReinhold2002}
\end{finite_members}

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\begin{subclasses}\ 

\href{Brouwerian_algebras.pdf}{Brouwerian algebras} 

\end{subclasses}

\begin{superclasses}\ 

\href{Semilattices_with_identity.pdf}{Semilattices with identity} 

\href{Hoops.pdf}{Hoops} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{ErneHeitzigReinhold2002}
M. Ern\'e, J. Heitzig, J. Reinhold,
\emph{On the number of distributive lattices}, 
Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.

\end{thebibliography}

\end{document}
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