Research PaperNo Access

From noncommutative diagrams to anti-elementary classes

Friedrich Wehrung

LMNO, CNRS UMR 6139, Département de Mathématiques, Université de Caen Normandie, 14032 Caen Cedex, France

E-mail Address: friedrich.wehrung01@unicaen.fr

https://doi.org/10.1142/S0219061321500112Cited by:5 (Source: Crossref)

Abstract

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form $ℒ_{\infty λ}$ . We prove that many naturally defined classes are anti-elementary, including the following:

the class of all lattices of finitely generated convex $ℓ$ -subgroups of members of any class of $ℓ$ -groups containing all Archimedean $ℓ$ -groups;
the class of all semilattices of finitely generated $ℓ$ -ideals of members of any nontrivial quasivariety of $ℓ$ -groups;
the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
the class of all semilattices of finitely generated two-sided ideals of rings;
the class of all semilattices of finitely generated submodules of modules;
the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
(assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$ -frame.

The main underlying principle is that under quite general conditions, for a functor $Φ : 𝒜 \to ℬ$ , if there exists a noncommutative diagram $\vec{D}$ of $𝒜$ , indexed by a common sort of poset called an almost join-semilattice, such that

$Φ {\vec{D}}^{I}$ is a commutative diagram for every set $I$ ,
$Φ \vec{D} ≇ Φ \vec{X}$ for any commutative diagram $\vec{X}$ in $𝒜$ ,

then the range of $Φ$ is anti-elementary.

AMSC: 18A30, 18A35, 03E05, 06A07, 06A12, 06C20, 06D22, 06D35, 06F20, 08C05, 08A30, 16E20, 16E50

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