Grassmann-Plücker relations and matroids with coefficients

Dress, Andreas; Wenzel, Walter

doi:10.1016/0001-8708(91)90036-7

Cited by 49 publications

(40 citation statements)

References 11 publications

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“…especially Dress (1986), Dress and Wenzel (1991). We point out that the fuzzy ring that belongs to "Valuated Matroids" is the same as that underlying "Tropical Geometry".…”

Section: Introductionmentioning

confidence: 87%

Algebraic, tropical, and fuzzy geometry

Dress

Wenzel

2011

Beitr Algebra Geom

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“…especially Dress (1986), Dress and Wenzel (1991). We point out that the fuzzy ring that belongs to "Valuated Matroids" is the same as that underlying "Tropical Geometry".…”

Section: Introductionmentioning

confidence: 87%

Algebraic, tropical, and fuzzy geometry

Dress

Wenzel

2011

Beitr Algebra Geom

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“…We mention that the essential contents of the above theorem have been derived in [8] in the study of "matroids with coefficients." Theorems 3.1 and 3.3 together say that base axioms and circuit axioms of a valuated matroid are cryptomorphically equivalent.…”

Section: Axioms Of a Valuated Matroidmentioning

confidence: 99%

“…Let X and Y be elements of satisfying X = Y . Since satisfies (VC3), we can assume that there exist B B ∈ and v v ∈ V such that X = X B v and Y = X B v , using the notation (8).…”

Section: Lemma 410mentioning

confidence: 99%

On Circuit Valuation of Matroids

Murota

Tamura

2001

Advances in Applied Mathematics

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The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of R ∪ −∞ -valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of R ∪ −∞ -valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

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“…For further examples of matroids with coefficients in a fuzzy ring K see [DW1,2] and [W2]. Now we turn to -matroids with coefficients.…”

Section: Preliminariesmentioning

confidence: 99%

“…[DW4], [DT]). In [DW1] it is shown that matroids with coefficients can be defined in terms of Grassmann-Plücker maps.…”

Section: Introductionmentioning

confidence: 99%

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

Wenzel

1999

Advances in Applied Mathematics

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The concept of a combinatorial W P U -geometry for a Coxeter group W , a subset P of its generating involutions and a subgroup U of W with P ⊆ U yields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even -matroids. The concept of a W P -matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes of W and P one gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial W P U -geometries and suggest two equivalent definitions of a W P Umatroid with coefficients in a fuzzy ring K. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that W P U -matroids with coefficients encompass matroids with coefficients and -matroids with coefficients.

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Grassmann-Plücker relations and matroids with coefficients

Cited by 49 publications

References 11 publications

Algebraic, tropical, and fuzzy geometry

Algebraic, tropical, and fuzzy geometry

On Circuit Valuation of Matroids

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

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