An algebra (in the sense of universal algebra) of type is called a BCI-algebra if, for any , it satisfies the following conditions. (Informally, we may read as "truth" and as " implies ".)
BCI-1
BCI-2
BCI-3
BCI-4
BCI-5
BCK algebra
A BCI-algebra is called a BCK-algebra if it
satisfies the following condition:
BCK-1
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:
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.