Repository landing page
From non-commutative diagrams to anti-elementary classes
Abstract
56 pages. The final version (sent to the publisher) is v4.International audienceAnti-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 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 K_0-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 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 is not isomorphic to Φ X for any commutative diagram X in A, then the range of Φ is anti-elementary- info:eu-repo/semantics/article
- Journal articles
- lifter
- category
- functor
- Armature Lemma
- lattice
- distributive
- Cevian
- lattice-ordered group
- ring
- nonstable K-theory
- coordinatization
- norm-covering
- colimit
- presented
- condensate
- directed
- elementary
- anti-elementary
- DCPO
- scaled
- Boolean algebra
- Accessible category
- uniformization
- commutative
- diagram
- 18A30; 18A35; 03E05; 06A07; 06A12; 06C20; 06D22; 06D35; 06F20; 08C05; 08A30; 16E20; 16E50
- [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
- [MATH.MATH-LO]Mathematics [math]/Logic [math.LO]