Stone duality for lattice expansions

Hartonas, Chrysafis

doi:10.1093/jigpal/jzy010

Cited by 20 publications

(16 citation statements)

References 22 publications

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“…In the case of interest in this article, the closure operator ∎ carries over the residuation structure of A to A • . More precisely, our representation results [37,39] have established the following: Theorem 1.1 ( [37,39]). Every residuated lattice L = (L, ∧, ∨, 0, 1, ←, ○, →) is a sublattice of the lattice A • of a sorted, residuated modal algebra ∶ A ⇆ B ∶ ∎ where, moreover, (A, , ⊙, ) is a residuated Boolean algebra and where the lattice operator ○ is the closure of the restriction of ⊙ on A • , while its residuals ←, → are simply the restrictions of the residuals , of ⊙ on A • .…”

Section: Introductionmentioning

confidence: 76%

“…then extended to all stable sets using join-density of closed elements by setting A ○ σ C = ⋁ z∈A,z ′ ∈C (Γz ○ σ Γz ′ ). As pointed out in [35][36][37]39], this delivers the same map on stable sets as the map defined on closed elements by setting Γz ◯ Γz ′ = Γ(z○z ′ ), then again extending to all stable sets by using join-density of closed elements. This is immediate since…”

Section: By Defn Of Modal Translationmentioning

confidence: 96%

“…Besides serving as a means to construct the σ-extension of maps, as shown in [39,41], point operators are used in [34] and in each of [39,41] to construct relational structures dual to lattice expansions. Roughly, for each normal lattice…”

Section: Canonical Extensionsmentioning

confidence: 99%

“…is then appropriately derived. The relations R f , R ∂ f are then used to define operators F on stable sets and F ∂ on co-stable sets that are shown in [39,41] to be the canonical (dual) extension f σ , f ∂ σ of f . We conclude this section with considering the case where the represented lattice (expansion) is distributive, or a Boolean algebra.…”

Section: Canonical Extensionsmentioning

confidence: 99%

“…Following [34] and as detailed in [37,39], define the canonical relation R 111 by first defining a point operator x○z = ⋁{xa○b a ∈ x, b ∈ z} on filters, where x, z, z ′ ∈ X and x e = e ↑ designates a principal filter and then letting xR 111 zz ′ iff z○z ′ ≤ x, where ≤ designates filter inclusion. Note that for any x, z, z ′ we easily obtain from the definition that Table 1.…”

Section: By Defn Of Modal Translationmentioning

confidence: 99%

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Modal translation of substructural logics

Hartonas

2020

Journal of Applied Non-Classical Logics

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In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalizing the wellknown Gödel-McKinsey-Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness, Löwenheim-Skolem property and decidability.intuitionistic formulas. Grzegorczyk [32] showed that adding to S4 the axiom (G) ◻(◻(p→◻p)→p)→p, the (now known as) Grzegorczyk system Grz = S4+G is strictly stronger than S4, it is not contained in S5 and the same result about the GMT-translation of IL into Grz is provable. This was also shown to be true for McKinsey's system S4.1 = S4+(◻ ϕ → ◻ ϕ). This quickly led to the investigation of logics, variably called super-intuitionistic, or intermediate logics between IL and CPL and their modal companions, i.e. modal logics containing S4, cf. [5,12,21,47,48], in some cases logics between S4 and S5 [16], which culminated in the Blok-Esakia theorem, independently proven by Blok [8] and Esakia [21]. The topological interpretation of S4 [49-51], interpreting the box operator as an interior operator, provided furthermore the background for investigating dualities between modal companions of superintuitionistic logics and topological spaces [5]. In all of the above cases distribution is assumed and topological dualities investigated are based on the Priestley [59] duality for distributive lattices, or on the Esakia [20-22] duality for Heyting algebras.For logics without distribution, Goldblatt [29] provided a translation of Orthologic O into the B system of modal logic. This quickly led to modal approaches to quantum logic [4,58], as an immediate application. Goldblatt's result is specific to Orthologic and it was Kosta Došen [15] who embarked in a project of extending the GMT translation to the case of substructural logics. Došen's main result is an embedding of BCI (to which he refers as BC), BCK, BCW with intuitionistic negation into S4-type modal extensions of versions of BCI, BCK, BCW, respectively, all equipped with a classical-type negation (satisfying double negation and De Morgan laws). The result is established by proof-theoretic means, but the approach does not apply uniformly to substructural logics at large. Indeed, Došen points out that difficulties arise when dealing with weak systems such as the non-commutative and/or non-associative full Lambek Calculus (N)FL (see [15], Introduction, as well as end of Section 4).A paralle...

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Section: Introductionmentioning

confidence: 76%

Section: By Defn Of Modal Translationmentioning

confidence: 96%

Section: Canonical Extensionsmentioning

confidence: 99%

Section: Canonical Extensionsmentioning

confidence: 99%

Section: By Defn Of Modal Translationmentioning

confidence: 99%

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Modal translation of substructural logics

Hartonas

2020

Journal of Applied Non-Classical Logics

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Duality for normal lattice expansions and sorted residuated frames with relations

Hartonas

2023

Algebra Univers.

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We revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.

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Choice-free topological duality for implicative lattices and Heyting algebras

Hartonas

2023

Algebra Univers.

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Stone duality for lattice expansions

Cited by 20 publications

References 22 publications

Modal translation of substructural logics

Modal translation of substructural logics

Duality for normal lattice expansions and sorted residuated frames with relations

Choice-free topological duality for implicative lattices and Heyting algebras

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