Mathematical Structures: BCK-meet-semilattices

# BCK-meet-semilattices

http://mathcs.chapman.edu/structuresold/files/BCK-meet-semilattices.pdf
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\begin{document}
\textbf{\Large BCK-meet-semilattices}

\abbreviation{BCKJMlat}

\begin{definition}
A \emph{BCK-meet-semilattice} is a structure $\mathbf{A}=\langle A,\wedge,\rightarrow,1\rangle$ 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):  $(x\wedge y)\rightarrow y = 1$

(5):  $x\wedge((x\rightarrow y)\rightarrow y) = x$

$\wedge$ is idempotent:  $x\wedge x = x$

$\wedge$ is commutative:  $x\wedge y = y\wedge x$

$\wedge$ is associative:  $(x\wedge y)\wedge z = x\wedge (y\wedge z)$

Remark:
$x\le y \iff x\rightarrow y=1$ is a partial order, with $1$ as greatest element, and $\wedge$ is a meet in this partial order. \cite{Idziak1984}
\end{definition}

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

$h(x\wedge y)=h(x)\wedge h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ 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 & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\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/}
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\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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