Elementary equivalences and accessible functors

Beke, Tibor; Rosický, Jiřı́

doi:10.1016/j.apal.2018.03.004

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…The following result is essentially a reformulation, in our context, of Feferman's [10,Theorem 6] or Beke and Rosický's [4,Proposition 2.14]. However, while the preservation of monomorphisms belongs to the assumptions of the two abovementioned results, it appears in our result as a conclusion (for monomorphisms with large enough domain).…”

Section: Purity and Freshnessmentioning

confidence: 57%

From noncommutative diagrams to anti-elementary classes

Wehrung

2020

J. Math. Log.

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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.

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Section: Purity and Freshnessmentioning

confidence: 57%

From noncommutative diagrams to anti-elementary classes

Wehrung

2020

J. Math. Log.

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“…The case where ¢ is function extension, f = f , d(f ) = dom f , and r(f ) = rng f whenever f ∈ F, is Dickmann's original p,e λ from [6, Definition 4.2.3]. As many aspects of the present paper are category-theoretical, we should point the reader to Beke and Rosický's categorical take on back-and-forth presented in [3].…”

Section: From Rpc To Pc In the Singular Casementioning

confidence: 95%

“…The main result of this section (i.e., Theorem 12.4) implies that DSLat 0 does not have all pushouts or, for that matter, pullbacks; hence it is not locally presentable. To that end, consider the Boolean semilattice 2 def = {0, 1} and define (∨, 0)-semilattice embeddings ι : 2 → 2 2, α, β : 2 2 → 3 2, and γ 3 , γ 4 : 3 Our next task will be representing DSLat 0 as the image of an accessible functor (by Theorem 5.5 this is certainly possible). Moreover, the domain of that functor will be constructed as a (finitary, congruence-distributive) variety of algebras.…”

Section: A Finite Bounded Poset For Which Cll Failsmentioning

confidence: 99%

“…In Section 12 we produce a counterexample for that problem. Roughly speaking, this counterexample consists of a faithful, finitely 3 The latter result would not follow from some hypothetical infinitary translation of Skolem normal form reduction a priori: for one thing, even the less demanding prenex normal form reduction may fail in L ∞λ (cf. Dickmann [6,Fact 1.1.1]).…”

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confidence: 99%

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Projective classes as images of accessible functors

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2022

Journal of Logic and Computation

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We are dealing with projective classes (in short $\textrm {PC}$) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$ for any infinite cardinal $\lambda $, and that if $\lambda $ is singular, then $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$. If $\lambda $ is regular, then a class of structures over a $\lambda $-ary vocabulary is $\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$-definable iff it is the image of a $\lambda $-continuous functor on a $\lambda $-accessible category; we also provide separating counterexamples for the non $\lambda $-ary case. We prove that many $\textrm {PC}$ classes of structures, previously known not to be closed under elementary equivalence over any $\mathbin {\mathscr {L}}_{\infty \lambda }$, are not even $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. Those classes arise from diverse contexts including convex $\ell $-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K$_0$-theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is $\textrm {PC}$ but not $\textrm {co}\textrm {-}\textrm {PC}$ over $\mathbin {\mathscr {L}}_{\infty \infty }$. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a ‘well-behaved’ functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.

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“…. Since directed colimits in Bool P are standard, 4 As established in [13, Proposition 2.3.2], the category Bool P has also all binary products.…”

Section: 4]) the Pair (Ultmentioning

confidence: 99%

From non-commutative diagrams to anti-elementary classes

Wehrung

2019

Preprint

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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.

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Elementary equivalences and accessible functors

Cited by 3 publications

References 18 publications

From noncommutative diagrams to anti-elementary classes

From noncommutative diagrams to anti-elementary classes

Projective classes as images of accessible functors

From non-commutative diagrams to anti-elementary classes

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