=====BCI-algebras=====

Abbreviation: **BCI**
====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):  $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$


(5):  $x\cdot 0 = 0 \Longrightarrow x=0$


Remark: 

==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: 

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

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |Quasivariety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |No |
^[[Residual size]]  | |
^[[Congruence distributive]]  |No |
^[[Congruence modular]]  |No |
^[[Congruence n-permutable]]  |No |
^[[Congruence regular]]  |No |
^[[Congruence uniform]]  |No |
^[[Congruence extension property]]  |No |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

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

====Subclasses====
[[BCK-algebras]] 

====Superclasses====
[[Groupoids]] 


====References====

[(Ln19xx>
)]