On the relation between hyperrings and fuzzy rings

Giansiracusa, Jeffrey; Jun, Jaiung; Lorscheid, Oliver

doi:10.1007/s13366-017-0347-5

Cited by 18 publications

(21 citation statements)

References 16 publications

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“…In their recent preprint [GJL16], Jeff Giansiracusa, Jaiung Jun, and Oliver Lorscheid show that there is a fully faithful functor from hyperfields to fuzzy rings which induces an equivalence between the theory of strong matroids over a hyperfield and the theory of matroids over the corresponding fuzzy ring. More precisely, their functor induces an equivalence of categories between hyperfields and field-like fuzzy rings which identifies strong matroids over the former with matroids over the latter.…”

Section: Introductionmentioning

confidence: 99%

Matroids over hyperfields

Baker

Bowler

2018

AIP Conference Proceedings

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We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids, as well as phased matroids in the sense of Anderson-Delucchi. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plücker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.

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Section: Introductionmentioning

confidence: 99%

Matroids over hyperfields

Baker

Bowler

2018

AIP Conference Proceedings

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“…The forward implication of part (a) was obtained decades earlier by Edmonds [12], of which it follows from Proposition 19 and the forms of the inequalities in (17). In part (b) we allow i = j.…”

Section: The Polyhedron P (M )mentioning

confidence: 77%

“…Comparison of matroid generalisations. Matroids over rings differ in an apparent way from the "orthodox" picture of generalised matroids which has flourished in the last years, represented by matroids over hyperfields [2], over fuzzy rings [10] (these last two having been unified in [17]), and over partial fields [30]. All of these objects set about generalising the coefficients in a realisation: so when the base is made to be a field K, what comes out are matroids realised over K, and to capture simply the set of all classical matroids, one needs a new base "simpler" than any field.…”

Section: Introductionmentioning

confidence: 99%

Polyhedra and parameter spaces for matroids over valuation rings

Fink¹,

Moci²

2019

Advances in Mathematics

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In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott-Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids.

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“…Dress [16] introduced "fuzzy rings" a while ago in connection with matroids, and these also have been seen recently to be related to hypergroups in [5,17,18,25,56].…”

Section: Fuzzy Ringsmentioning

confidence: 99%