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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>
 +)]
 +
 +
 +
 +