Priestley duality for MV-algebras and beyond

Fussner, Wesley; Gehrke, Mai; Gool, Samuel J. van; Marra, Vincenzo

doi:10.1515/forum-2020-0115

Cited by 5 publications

(5 citation statements)

References 26 publications

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“…Next, we show that ∼ is second-order-definable in terms of −. The proof follows the same ideas as the proof of Proposition 3.14 in [6].…”

Section: The First Partial Operation Represents the Pi Extension Of O...mentioning

confidence: 69%

“…Traditionally, given a distributive lattice with a quasioperator, one endows the dual Priestley space with two ternary relations satisfying certain topological and order-theoretic properties which uniquely characterise the quasioperator. As was already evident in [12,13] (and successfully applied in [6]), these two relations arise from two partial operations, by taking the upwards or downwards closures of the codomains.…”

Section: Discussionmentioning

confidence: 89%

“…For example, if A comes equipped with a binary operation h : A × A → A then, by studying the relationship between the graph of h and the points of X in the product A δ × A δ × A δ , we can identify two partial binary operations on X that fully capture h. An instance of this situation is studied in this section. Our main inspiration comes from [12,13] and also [6].…”

Section: Appendix a Extended Priestley Duality And Canonical Extensionsmentioning

confidence: 99%

“…In fact, A δ is isomorphic to the lattice of all up-sets of the Priestley space dual to A. 6 In the following, we will always identify A with a sublattice of its canonical extension A δ . Open elements of A δ are those of the form S, for some S ⊆ A.…”

Section: Appendix a Extended Priestley Duality And Canonical Extensionsmentioning

confidence: 99%

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A duality theoretic view on limits of finite structures: Extended version

Gehrke,

Jakl,

Reggio

2020

Preprint

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A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises-via Stone-Priestley duality and the notion of types from model theory-by enriching the expressive power of first-order logic with certain "probabilistic operators". We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction.The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.

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“…Next, we show that ∼ is second-order-definable in terms of −. The proof follows the same ideas as the proof of Proposition 3.14 in [6].…”

Section: The First Partial Operation Represents the Pi Extension Of O...mentioning

confidence: 69%

Section: Discussionmentioning

confidence: 89%

Section: Appendix a Extended Priestley Duality And Canonical Extensionsmentioning

confidence: 99%

Section: Appendix a Extended Priestley Duality And Canonical Extensionsmentioning

confidence: 99%

See 2 more Smart Citations

A duality theoretic view on limits of finite structures: Extended version

Gehrke,

Jakl,

Reggio

2020

Preprint

Self Cite

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“…In [25] we find a duality between a generalization of MV-algebras and Priestley spaces with an appropriate extra structure. Note that the Priestley space of a distributive lattice can be realized in various ways, for instance we can take the set of the prime ideals of the lattice, equipped with the inclusion order and with the patch topology (not with the Zariski topology).…”

Section: Introductionmentioning

confidence: 87%

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

Barbieri¹,

Nola²,

Lenzi³

2023

Preprint

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As a main result, we characterize prime spectra of abelian lattice ordered groups. Further we introduce some categories based on spectral spaces, lattices and Priestley spaces, and we relate these categories with each other and with the category of presented MV-algebras, by means of functors. We turn to lattices and offer a simple characterization of 1) maps whose Stone dual preserves closed sets, and 2) closed epimorphisms between distributive lattices as well as their Stone duals. We have a characterization of the variety generated by the Chang MV-algebra and we study this variety. Next we generalize the results to every variety generated by a Komori chain. Finally we discuss homogeneous polynomials in MV-algebras.

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Topological Duality and Algebraic Completions

Gehrke

2021

Outstanding Contributions to Logic

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Priestley duality for MV-algebras and beyond

Cited by 5 publications

References 26 publications

A duality theoretic view on limits of finite structures: Extended version

A duality theoretic view on limits of finite structures: Extended version

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

Topological Duality and Algebraic Completions

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