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Output written on BCK-join-semilattices.pdf (2 pages, 69853 bytes).
Transcript written on BCK-join-semilattices.log.
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\newtheorem*{morphisms}{Morphisms}
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\newtheorem*{finite_members}{Finite Members}
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\begin{document}
\textbf{\Large BCK-join-semilattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=BCK-join-semilattices}{edit}
\abbreviation{BCKJSlat}
\begin{definition}
A \emph{BCK-join-semilattice} is a structure $\mathbf{A}=\langle A,\vee,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that
(1): $(x\rightarrow y)\rightarrow ((y\rightarrow z)\rightarrow (x\rightarrow z)) = 1$
(2): $1\rightarrow x = x$
(3): $x\rightarrow 1 = 1$
(4): $x\rightarrow (x\vee y) = 1$
(5): $x\vee((x\rightarrow y)\rightarrow y) = ((x\rightarrow y)\rightarrow y)$
$\vee$ is idempotent: $x\vee x = x$
$\vee$ is commutative: $x\vee y = y\vee x$
$\vee$ is associative: $(x\vee y)\vee z = x\vee (y\vee z)$
Remark:
$x\le y \iff x\rightarrow y=1$ is a partial order, with $1$ as greatest element, and $\vee$ is a join
for this order. \cite{Idziak1984}
\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-join-semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ and $h(1)=1$
\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 & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & \\\hline
Residual size & \\\hline
Congruence distributive & \\\hline
Congruence modular & \\\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 $n$.
$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\
\href{BCK-lattices.pdf}{BCK-lattices}
\end{subclasses}
\begin{superclasses}\
\href{BCK-algebras.pdf}{BCK-algebras}
\end{superclasses}
\begin{thebibliography}{10}
\bibitem{Idziak1984}
Pawel M. Idziak, \emph{Lattice operation in BCK-algebras},
Math. Japon., \textbf{29}, 1984, 839--846 \href{http://www.ams.org/mathscinet-getitem?mr=87b:06025a}{MRreview}
\end{thebibliography}
\end{document}
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