Geodesic
From Specker ℓ-groups to boolean algebras via Γ
Theory and applications of categories, Tome 41 (2024), pp. 825-837

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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.

Publié le :
Classification : Primary: 06F20, 06D35, Secondary: 06E05, 18F60, 18F70, 46L80, 47L40
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 
@article{TAC_2024_41_a24,
     author = {Daniele Mundici},
     title = {From {Specker} \ensuremath{\ell}-groups to boolean algebras via {\ensuremath{\Gamma}}},
     journal = {Theory and applications of categories},
     pages = {825--837},
     publisher = {mathdoc},
     volume = {41},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a24/}
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Daniele Mundici. From Specker ℓ-groups to boolean algebras via Γ. Theory and applications of categories, Tome 41 (2024), pp. 825-837. http://geodesic.mathdoc.fr/item/TAC_2024_41_a24/