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From non-commutative diagrams to anti-elementary classes
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 . We prove
that many naturally defined classes are anti-elementary, including the
following: the class of all lattices of finitely generated convex
{\ell}-subgroups of members of any class of {\ell}-groups containing all
Archimedean {\ell}-groups; the class of all semilattices of finitely
generated {\ell}-ideals of members of any nontrivial quasivariety of
{\ell}-groups; the class of all Stone duals of spectra of
MV-algebras-this yields a negative solution for 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 -theory of von Neumann regular rings, respectively C*-algebras
of real rank zero; (assuming arbitrarily large Erd"os 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 : 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 D^I is a commutative
diagram for every set I, D is not isomorphic to X for
any commutative diagram X in A, then the range of is anti-elementary.Comment: 49 pages. Journal of Mathematical Logic, World Scientific Publishing,
In pres