Completely representable lattices

Egrot, Rob; Hirsch, Robin

doi:10.1007/s00012-012-0181-4

Cited by 9 publications

(11 citation statements)

References 9 publications

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“…Theorem 5.1 below, which also appears as [6, theorem 3.2], shows that the extra structure of Boolean algebras is necessary here as the class of completely representable posets (distributive lattices to be exact) is not closed under elementary equivalence and hence cannot be elementary. It is conjectured in [6] that the classes of (ω, C) and (C, ω)-representable distributive lattices are not elementary. We conjecture here that the classes of (α, C) and (C, β)-representable posets are not elementary for all α, β ≥ 3.…”

Section: Complete Representations For Posetsmentioning

confidence: 99%

Representable posets

Egrot

2016

Journal of Applied Logic

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A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals α and β a poset is said to be (α, β)-representable if an embedding into a field of sets exists that preserves meets of sets smaller than α and joins of sets smaller than β. We show using an ultraproduct/ultraroot argument that when 2 ≤ α, β ≤ ω the class of (α, β)-representable posets is elementary, but does not have a finite axiomatization in the case where either α or β = ω. We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.

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Section: Complete Representations For Posetsmentioning

confidence: 99%

Representable posets

Egrot

2016

Journal of Applied Logic

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“…Complete representations have been studied previously in the context of various different forms of representability: by sets [7], by binary or higher-order relations [12,13], or by partial functions [23]. Definition 2.8.…”

Section: Algebras Of Functionsmentioning

confidence: 99%

Difference-restriction algebras of partial functions with operators: discrete duality and completion

Borlido,

McLean

2020

Preprint

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We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.

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“…A partially ordered set (poset) is representable if it can be embedded into a powerset algebra via a map that preserves existing finite meets and joins. The class of representable posets and its infinitary variations have been studied, not always using this terminology, in [8, 11–15, 21, 29, 39], generalising work done in the setting of semilattices [2, 9, 27, 33], and for distributive lattices and Boolean algebras [1, 3, 4, 6, 7, 16, 31, 35, 36]. At first glance, it is far from obvious that the class of representable posets is elementary.…”

Section: Introductionmentioning

confidence: 99%

Recursive Axiomatisations From Separation Properties

Egrot¹

2021

J. symb. log.

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We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.

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Completely representable lattices

Cited by 9 publications

References 9 publications

Representable posets

Representable posets

Difference-restriction algebras of partial functions with operators: discrete duality and completion

Recursive Axiomatisations From Separation Properties

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