From noncommutative diagrams to anti-elementary classes

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 ∞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 K 0-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 → B, if there exists a non-commutative diagram D of A, indexed by a common sort of poset called an almost join-semilattice, such that • Φ D I is a commutative diagram for every set I, • Φ D ∼ = Φ X for any commutative diagram X in A, then the range of Φ is anti-elementary.

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

“…The papers [41] and [42] contain a deep study of the spectrum problem from a logical point of view. In [41] it is shown that second countable spectral spaces which are prime spectra of MV-algebras are characterized by a simple first order formula, whereas in [42] is is shown that no infinitary first order logic can express the prime spectra of MV-algebras. Other information is contained in [43].…”