#
From Specker ℓ-groups to boolean algebras via Γ

##
Daniele Mundici

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.

http://www.tac.mta.ca/tac/volumes/41/25/41-25.pdf

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