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bck-algebras [2010/07/29 15:23] (current)
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+=====BCK-algebras=====
+
+Abbreviation: **BCK**
+====Definition====
+A \emph{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):  $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$
+
+Remark:
+$x\le y \iff x\cdot y=0$ is a partial order, with $0$ as least element.
+
+BCK-algebras provide [[algebraic semantics]] for BCK-logic, named after
+the combinators B, C, and K by C. A. Meredith, see [(Prior1962)].
+
+====Definition====
+A \emph{BCK-algebra} is a [[BCI-algebra]]
+$\mathbf{A}=\langle A,\cdot ,0\rangle$ such that
+
+$x\cdot 0 = x$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+$h(x\cdot y)=h(x)\cdot h(y)$ and $h(0)=0$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]                        |quasivariety [(Wronski1983)] |
+^[[Equational theory]]                | |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               |undecidable |
+^[[Locally finite]]                   |no |
+^[[Residual size]]                    |unbounded |
+^[[Congruence distributive]]          |no |
+^[[Congruence modular]]               |no |
+^[[Congruence n-permutable]]          |no |
+^[[Congruence regular]]               |no |
+^[[Congruence uniform]]               |no |
+^[[Congruence extension property]]    |no |
+^[[Definable principal congruences]]  |no |
+^[[Equationally def. pr. cong.]]      |no |
+^[[Amalgamation property]]            |yes |
+^[[Strong amalgamation property]]     |yes [(Wronski1984)] |
+^[[Epimorphisms are surjective]]      | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+[[Commutative BCK-algebras]]
+
+====Superclasses====
+[[BCI-algebras]]
+
+
+====References====
+
+[(Prior1962>
+A. N. Prior, \emph{Formal logic},
+Second edition, Clarendon Press, Oxford, 1962, p.316
+[[MRreview]]
+
+[(Wronski1983>
+Andrzej Wronski,\emph{BCK-algebras do not form a variety},
+Math. Japon., \textbf{28}, 1983, 211--213 [[http://www.ams.org/mathscinet-getitem?mr=84e:06015|MRreview]]
+
+[(Wronski1984>
+Andrzej Wronski,\emph{Interpolation and amalgamation properties of BCK-algebras},
+Math. Japon., \textbf{29}, 1984, 115--121 [[http://www.ams.org/mathscinet-getitem?mr=85e:06015|MRreview]]
+)]
+
+
+
+