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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> | ||
+ | )] | ||
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+ | |||
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