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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>
		+	)]
		+
		+
		+
		+