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 [Formula: see text]. We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex [Formula: see text]-subgroups of members of any class of [Formula: see text]-groups containing all Archimedean [Formula: see text]-groups; the class of all semilattices of finitely generated [Formula: see text]-ideals of members of any nontrivial quasivariety of [Formula: see text]-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 [Formula: see text]-frame. The main underlying principle is that under quite general conditions, for a functor [Formula: see text], if there exists a noncommutative diagram [Formula: see text] of [Formula: see text], indexed by a common sort of poset called an almost join-semilattice, such that [Formula: see text] is a commutative diagram for every set [Formula: see text], [Formula: see text] for any commutative diagram [Formula: see text] in [Formula: see text], then the range of [Formula: see text] is anti-elementary.