BCK algebra

Definition

BCI algebra

An algebra (in the sense of universal 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. (Informally, we may read 0 as "truth" and x\ast y as "y implies x".) ; 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 \land y\ast x=0\implies x=y ; BCI-5: x\ast 0=0 \implies x=0

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.

A partial order can then be defined as x ≤ y iff x * y = 0.

A BCK-algebra is said to be commutative if it satisfies: : x\ast (x\ast y)= y\ast (y\ast x) In a commutative BCK-algebra x * (x * y) = x ∧ y is the greatest lower bound of x and y under the partial order ≤.

A BCK-algebra is said to be bounded if it has a largest element, usually denoted by 1. In a bounded commutative BCK-algebra the least upper bound of two elements satisfies x ∨ y = 1 * ((1 * x) ∧ (1 * y)); that makes it a distributive lattice.

Examples

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

The subsets of a set form a BCK-algebra, where A*B is the difference A\B (the 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).

The bounded commutative BCK-algebras are precisely the MV-algebras.

References

• {{citation|title=Review of several papers on BCI, BCK-Algebras
• {{citation|url=http://projecteuclid.org/euclid.pja/1195522126
• Y. Huang, BCI-algebra, Science Press, Beijing, 2006.