A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

Marti, Johannes; Pinosio, Riccardo

doi:10.1007/s11083-019-09497-0

Cited by 6 publications

(28 citation statements)

References 35 publications

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“…8. There are some definitions of infinite convex geometries in the literature: see, for instance, Adaricheva (2014b), Adaricheva andNation (2016d), Jamison-Waldner (1982), Mao (2017), Mao and Liu (2012), Marti and Pinosio (2020), Wahl (2001). In view of these notions, extending the investigation of resolutions to infinite convex geometries appears to be of some interest.…”

Section: Future Work and Open Problemsmentioning

confidence: 99%

Resolutions of Convex Geometries

Cantone

Doignon

Giarlotta

et al. 2021

Electron. J. Combin.

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Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions–compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Möhring and Radermacher)–, resolutions of convex geometries always yield a convex geometry. We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.

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Section: Future Work and Open Problemsmentioning

confidence: 99%

Resolutions of Convex Geometries

Cantone

Doignon

Giarlotta

et al. 2021

Electron. J. Combin.

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“…It can be shown that G * is a closure operator. Moreover, for a Plott function G, the closure G * (X) is a kind of convex hull of X (for details see [10] for finite C and [13] in the general case). The inversion to the operator G * is defined by the formula:…”

Section: Closure Operatormentioning

confidence: 99%

Stable sets of contracts in two-sided markets

Danilov,

Koshevoy

2021

Preprint

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We revisit the problem of existence of stable systems of contracts with arbitrary sets of contracts. We show that stable sets of contracts exists if choices of agents satisfy path-independence. We call such choice functions Plott functions. Our proof is based on application of Zorn lemma to a special poset of semi-stable pairs. Moreover, we construct a dynamic process on the poset (generalizing algorithm Gale and Shapley) steady states of which are stable sets.In Appendix we discuss Lehmann hyper-orders and establish a bijection between the set of Lehmann hyper-orders and the set of Plott functions.

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“…8. There are some definitions of infinite convex geometries in the literature: see, for instance, Adaricheva (2014b), Adaricheva and Nation (2016), Jamison-Waldner (1982), Mao (2017), Mao and Liu (2012), Marti andPinosio (2020), Wahl (2001). In view of these notions, extending the investigation of resolutions to infinite convex geometries appears to be of some interest.…”

Section: Future Work and Open Problemsmentioning

confidence: 99%

Resolutions of Convex Geometries

Cantone¹,

Doignon²,

Giarlotta³

et al. 2021

Preprint

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Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions-compounds of hypergraphs, as in Chein, Habib, and Maurer (1981), and compositions of set systems, as in Möhring and Radermacher (1984)-, resolutions of convex geometries always yield a convex geometry.We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.

show abstract

A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

Cited by 6 publications

References 35 publications

Resolutions of Convex Geometries

Resolutions of Convex Geometries

Stable sets of contracts in two-sided markets

Resolutions of Convex Geometries

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