History of BCI-algebras

 
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 \begin{document}
 \textbf{\Large BCI-algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=BCI-algebras}{edit}
 
 \abbreviation{BCI}
 \begin{definition}
 A \emph{BCI-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):  $(x\cdot (x\cdot y))\cdot y = 0$
 
 
 (3):  $x\cdot x = 0$
 
 
 (4):  $x\cdot y=y\cdot x= 0 \implies x=y$
 
 
 (5):  $x\cdot 0 = 0 \implies x=0$
 
 
 Remark: 
 
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be BCI-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\\\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 & \\\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{BCK-algebras.pdf}{BCK-algebras} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Groupoids.pdf}{Groupoids} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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