# Differences

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bci-algebras [2010/07/29 15:23] (current)
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+=====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>
+)]
+
+
+
+