Commutative BCK-algebras

http://mathcs.chapman.edu/structuresold/files/Commutative_BCK-algebras.pdf

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

\abbreviation{ComBCK}
\begin{definition}
A \emph{commutative 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):  $x\cdot y=y\cdot x= 0 \implies x=y$


(5):  $x\cdot (x\cdot y) = y\cdot (y\cdot x)$

Remark: 
Note that the commutativity does not refer to the operation $\cdot$, but rather to the
term operation $x\wedge y=x\cdot (x\cdot y)$, which turns out to be a meet with respect
to the following partial order:

$x\le y \iff x\cdot y=0$, with $0$ as least element.

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

$x\cdot (x\cdot y) = y\cdot (y\cdot x)$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be commutative 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                       & variety\\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & no\\\hline
Residual size                   & unbounded\\\hline
Congruence distributive         & yes\\\hline
Congruence modular              & yes\\\hline
Congruence n-permutable         & yes, $n=3$\\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\hline
Congruence extension property   & \\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong.     & no\\\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 $n$.

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\begin{subclasses}\ 

\href{Tarski_algebras.pdf}{Tarski algebras} 

\end{subclasses}
\begin{superclasses}\ 

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

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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