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commutative_bck-algebras [2010/07/29 15:46] (current)
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+=====Commutative BCK-algebras=====
+
+Abbreviation: **ComBCK**
+====Definition====
+A \emph{commutative BCK-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):  $x\cdot 0 = x$
+
+
+(3):  $0\cdot x = 0$
+
+
+(4):  $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$
+
+
+(5):  $x\cdot (x\cdot y) = y\cdot (y\cdot x)$
+
+Remark:
+Note that the commutativity does not refer to the operation $\cdot$, but rather to the
+term operation $x\wedge y=x\cdot (x\cdot y)$, which turns out to be a meet with respect
+to the following partial order:
+
+$x\le y \iff x\cdot y=0$, with $0$ as least element.
+
+====Definition====
+A \emph{commutative BCK-algebra} is a [[BCK-algebra]]
+$\mathbf{A}=\langle A,\cdot ,0\rangle$ such that
+
+$x\cdot (x\cdot y) = y\cdot (y\cdot x)$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be commutative BCK-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]]                        |variety |
+^[[Equational theory]]                | |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               | |
+^[[Locally finite]]                   |no |
+^[[Residual size]]                    |unbounded |
+^[[Congruence distributive]]          |yes |
+^[[Congruence modular]]               |yes |
+^[[Congruence n-permutable]]          |yes, $n=3$ |
+^[[Congruence regular]]               | |
+^[[Congruence uniform]]               | |
+^[[Congruence extension property]]    | |
+^[[Definable principal congruences]]  |no |
+^[[Equationally def. pr. cong.]]      |no |
+^[[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====
+[[Tarski algebras]]
+
+====Superclasses====
+[[BCK-algebras]]
+
+
+====References====
+
+[(Ln19xx>
+)]
+
+
+
+
+