Mathematical Structures: BCK-algebras

# BCK-algebras

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

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

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

$x\cdot 0 = x$

Remark:

\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:

$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(0)=0$

\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
Andrzej Wroński,\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}\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & No\\\hline
Residual size & \\\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 & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & Yes\\\hline
Strong amalgamation property & Yes
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}\\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
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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{Ln19xx}

\end{thebibliography}

\end{document}
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