BCK algebra

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BCI and BCK algebras are mathematical structures used to study parts of logic that involve implication. They were introduced in 1966 by Y. Imai, K. Iséki and S. Tanaka. An algebra consists of a set X, a binary operation * and a distinguished element 0. A BCI-algebra is a structure (X, *, 0) that satisfies axioms describing implication-like rules. A BCK-algebra is defined similarly, and from it we can define a partial order by saying x ≤ y if x * y = 0.

If a BCK-algebra is commutative, then x * (x * y) equals the meet x ∧ y (the greatest lower bound) under this order. If there is a largest element 1, the algebra is bounded. In a bounded, commutative BCK-algebra the join x ∨ y can be written as 1 * ((1 * x) ∧ (1 * y)), making the structure a distributive lattice.

Examples: any abelian group becomes a BCI-algebra if we set a * b = a − b and 0 as the identity. The subsets of a set form a BCK-algebra with A * B = A \ B and 0 = ∅. If we take a Boolean algebra and define A * B as A ∧ ¬B, that is a BCK-algebra. Finally, bounded commutative BCK-algebras are exactly MV-algebras.