BCK algebra

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In mathematics, BCI and BCK algebras are algebraic structures, introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics.

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[edit] Definition

[edit] BCI algebra

An algebra \left( X;\ast ,0\right) of type \left( 2,0\right) is called a BCI-algebra if, for any x,y,z\in X, it satisfies the following conditions:

BCI-1
\left( \left( x\ast y\right) \ast \left( x\ast z\right) \right) \ast \left( z\ast y\right) =0
BCI-2
\left( x\ast \left( x\ast y\right) \right) \ast y=0
BCI-3
x\ast x=0
BCI-4
x\ast y=0 \and y\ast x=0\implies x=y
BCI-5
x\ast 0=0 \implies x=0

[edit] BCK algebra

A BCI-algebra \left( X;\ast ,0\right) is called a BCK-algebra if it satisfies the following condition:

BCK-1
\forall x\in X: 0\ast x=0

[edit] Examples

Every abelian group of is a BCI-algebra, with * group subtraction and 0 the group identity.

The subsets of a set form a BCK-algebra, where A*B is the difference A\B (elements in A but not in B), and 0 is the empty set.

A Boolean algebra is a BCK algebra if A*B is defined to be A∧¬B (A does not imply B).

[edit] References