BCK-algebras

http://math.chapman.edu/structuresold/files/BCK-algebras.pdf

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

\abbreviation{BCK}
\begin{definition}
A \emph{BCK-algebra} is a structure $\mathbf{A}=\langle A,\cdot ,0\rangle$ of type $\langle 2,0\rangle$ such that

(1):  $((x\cdot y)\cdot (x\cdot z))\cdot (z\cdot y) = 0$

(2):  $x\cdot 0 = x$

(3):  $0\cdot x = 0$

(4):  $x\cdot y=y\cdot x= 0 \implies x=y$

Remark: 
$x\le y \iff x\cdot y=0$ is a partial order, with $0$ as least element.

BCK-algebras provide \href{Algebraic_semantics.pdf}{algebraic semantics} for BCK-logic, named after
the combinators B, C, and K by C. A. Meredith, see \cite{Prior1962}.
\end{definition}

\begin{definition}
A \emph{BCK-algebra} is a \href{BCI-algebras.pdf}{BCI-algebra} 
$\mathbf{A}=\langle A,\cdot ,0\rangle$ such that

$x\cdot 0 = x$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x\cdot y)=h(x)\cdot h(y)$ and $h(0)=0$
\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                       & quasivariety \cite{Wronski1983}\\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & undecidable\\\hline
Locally finite                  & no\\\hline
Residual size                   & unbounded\\\hline
Congruence distributive         & no\\\hline
Congruence modular              & no\\\hline
Congruence n-permutable         & no\\\hline
Congruence regular              & no\\\hline
Congruence uniform              & no\\\hline
Congruence extension property   & no\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong.     & no\\\hline
Amalgamation property           & yes\\\hline
Strong amalgamation property    & yes \cite{Wronski1984}\\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
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\end{finite_members}
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\begin{subclasses}\ 

\href{Commutative_BCK-algebras.pdf}{Commutative BCK-algebras} 

\end{subclasses}
\begin{superclasses}\ 

\href{BCI-algebras.pdf}{BCI-algebras} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Prior1962}
A. N. Prior, \emph{Formal logic},
Second edition, Clarendon Press, Oxford, 1962, p.316
\href{http://www.ams.org/mathscinet-getitem?mr=24:A1815}{MRreview}

\bibitem{Wronski1983}
Andrzej Wronski,\emph{BCK-algebras do not form a variety},
Math. Japon., \textbf{28}, 1983, 211--213 \href{"http://www.ams.org/mathscinet-getitem?mr=84e:06015"}{MRreview}

\bibitem{Wronski1984}
Andrzej Wronski,\emph{Interpolation and amalgamation properties of BCK-algebras},
Math. Japon., \textbf{29}, 1984, 115--121 \href{"http://www.ams.org/mathscinet-getitem?mr=85e:06015"}{MRreview}

\end{thebibliography}

\end{document}
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