On Categorical Equivalences of Commutative BCK-algebras |
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Authors: | Dvurečenskij Anatolij |
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Affiliation: | (1) Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia |
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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). This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | Commutative BCK-algebra with the relative cancellation property lattice ordered group universal group categorical equivalence MV-algebra conical algebra property (S) |
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