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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]]
 +)]
 +
 +
 +
 +