On Categorical Equivalences of Commutative BCK-algebras

Abstract

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras).