The author constructed in 1986 an equivalence Γ between abelian ℓ-groups with a strong unit and C.C. Chang MV-algebras. In 1958 Chang proved that boolean algebras coincide with MV-algebras satisfying the equation x ⊕ x = x. In this paper it is proved that Γ yields, by restriction, an equivalence between the category S of Specker ℓ-groups whose distinguished unit is singular, and the category of boolean algebras. As a consequence, Grothendieck's K_0 functor yields an equivalence between abelian Bratteli AF-algebras and the countable fragment of S. An equivalence in the opposite direction is obtained by a combination of Γ with the Stone and Gelfand dualities.
Keywords: The categorical equivalence Γ; ℓ-group; singular element; Specker ℓ-group; MV-algebra; boolean algebra; spectral space; AF-algebra; Gelfand duality; Grothendieck K_0
2020 MSC: Primary: 06F20, 06D35; Secondary: 06E05, 18F60, 18F70, 46L80, 47L40
Theory and Applications of Categories, Vol. 41, 2024, No. 25, pp 825-837.
Published 2024-07-25.
TAC Home