Uniform interpolation and compact congruences

Gool, Samuel J. van; Metcalfe, George; Tsinakis, Constantine

doi:10.1016/j.apal.2017.05.001

Cited by 20 publications

(34 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Following [12], we say that V admits right uniform deductive interpolation if Π(y) in the preceding condition is required to be finite.…”

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

“…Below we establish a useful technical result, proved in a slightly different form as Lemma 3.9 in [12].…”

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

“…Clearly, every locally finite variety is coherent. Less obviously, the property holds for the varieties of Heyting algebras (the main content of Pitts' theorem for IPC [25]), abelian groups, lattice-ordered abelian groups, and MV-algebras (see [12]). On the other hand, by Higman's embedding theorem, the variety of groups is not coherent, since there exists a finitely generated recursively presented group that is not finitely presented.…”

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

See 2 more Smart Citations

Uniform interpolation and coherence

Kowalski

Metcalfe

2019

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices.

show abstract

“…Following [12], we say that V admits right uniform deductive interpolation if Π(y) in the preceding condition is required to be finite.…”

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

“…Below we establish a useful technical result, proved in a slightly different form as Lemma 3.9 in [12].…”

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

Section: Uniform Deductive Interpolation and Coherencementioning

confidence: 99%

See 1 more Smart Citation

Uniform interpolation and coherence

Kowalski

Metcalfe

2019

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…Related forms of "consequence-based" interpolation (like "deductive interpolation") have been studied in detail in several works [17,31,8,39,30,10,9,24,16] (see in particular [24] for proofs and references for the connections between deductive interpolation and amalgamation through congruence extension property). For normal modal logics, in view of the above deduction theorem, it is easy to see that the local interpolation property implies the global one (but it is not equivalent to it, see [21]).…”

Section: Interpolation In Propositional Modal Logicmentioning

confidence: 99%

Modularity results for interpolation, amalgamation and superamalgamation

Ghilardi

Gianola

2018

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

Wolter in [38] proved that the Craig interpolation property transfers to fusion of normal modal logics. It is well-known [21] that for such logics Craig interpolation corresponds to an algebraic property called superamalgamability. In this paper, we develop model-theoretic techniques at the level of first-order theories in order to obtain general combination results transferring quantifier-free interpolation to unions of theories over non-disjoint signatures. Such results, once applied to equational theories sharing a common Boolean reduct, can be used to prove that superamalgamability is modular also in the non-normal case. We also state that, in such non-normal context, superamalgamability corresponds to a strong form of interpolation that we call "comprehensive interpolation property" (which consequently transfers to fusions).

show abstract

“…Proof. By the Craig interpolation theorem for IPC, it suffices to prove the statement for any formula ψ whose variables are contained in p (cf., e.g., [10,Prop. 3.5]).…”

Section: Theorem 2 Every Continuous P-morphism Between Finitely Coprmentioning

confidence: 99%

An open mapping theorem for finitely copresented Esakia spaces

Gool

Reggio

2018

Topology and its Applications

View full text Add to dashboard Cite

We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, self-contained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras.

show abstract

Uniform interpolation and compact congruences

Cited by 20 publications

References 22 publications

Uniform interpolation and coherence

Uniform interpolation and coherence

Modularity results for interpolation, amalgamation and superamalgamation

An open mapping theorem for finitely copresented Esakia spaces

Contact Info

Product

Resources

About