Matroids over hyperfields

Baker, Matt; Bowler, Nathan

doi:10.1063/1.5043953

Cited by 28 publications

(57 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we review basic definitions and properties for matroids over hyperfields first introduced by Baker and Bowler in [5]. Let's first recall the definition of a hyperfield.…”

Section: Matroids Over Hyperfieldsmentioning

confidence: 99%

“…Constructing a Grassmann-Plücker function from circuits is more difficult to describe, and requires the additional notion of dual pairs. An explicit description of this construction is unnecessary for our purposes; the interested reader is referred to [5].…”

Section: Matroids Over Hyperfieldsmentioning

confidence: 99%

“…The Hopf algebra associated to a set of isomorphism classes of matroids closed under taking minors and direct sums is called a matroid-minor Hopf algebra. In this paper, we generalize the construction of the matroid-minor Hopf algebra to the setting of matroids over hyperfields, first introduced by M. Baker and N. Bowler in [5]. Remark 1.1.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hopf algebras for matroids over hyperfields

2020

View full text Add to dashboard Cite

Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. Date: December 27, 2017. 2010 Mathematics Subject Classification. 05E99(primary), 16T05(secondary). Key words and phrases. matroid, hyperfield, matroid over hyperfields, Hopf algebra, minor, direct sum.In this sense B and C B carry the same information as C B and B C respectively; these are thus said to determine the same matroid on E "cryptomorphically."Example 2.2. The motivating examples of matroids (hinted at above) are given as follows:(1) Let V be a finite dimensional vector space and E ⊆ V a spanning set of vectors.The bases of V contained in E form the bases of a matroid on E, and the minimal dependent subsets of E form the circuits of a matroid on E. Furthermore, these are the same matroid.(2) Let Γ be a finite, undirected graph with edge set E (loops and parallel edges are allowed). The sets of edges of spanning forests in Γ form the bases of a matroid on E, and the sets of edges of cycles form the circuits of a matroid on E. Furthermore, these are the same matroid (called the graphic matroid of Γ).One can define the notion of isomorphisms of matroids as follows.Definition 2.3. Let M 1 (resp. M 2 ) be a matroid on E 1 (resp. E 2 ) defined by a set B 1 (resp. B 2 ) of bases. We say that M 1 is isomorphic to M 2 if there exists a bijection f :In this case, f is said to be an isomorphism.Example 2.4. Let Γ 1 and Γ 2 be finite graphs and M 1 and M 2 be the corresponding graphic matroids. Every graph isomorphism between Γ 1 and Γ 2 gives rise to a matroid isomorphism between M 1 and M 2 , but the converse need not hold.Recall that given any base B ∈ B(M ) and any element e ∈ E \ B, there is a unique circuit (fundamental circuit ) C B,e of e with respect to B such that C B,e ⊆ B ∪ {e}.One can construct new matroids from given matroids as follows:Definition 2.5 (Direct sum of matroids). Let M 1 and M 2 be matroids on E 1 and E 2 given by bases B 1 and B 2 respectively. The direct sum M 1 ⊕ M 2 is the matroid on E 1 ⊔ E 2 given by the bases B = {B 1 ⊔ B 2 | B i ∈ B i for i = 1, 2}.Remark 2.6. One can easily check that M 1 ⊕ M 2 is indeed a matroid on E 1 ⊔ E 2 .Definition 2.7 (Dual, Restriction, Deletion, and Contraction). Let M be a matroid on a finite set E M with the set B M of bases and the set C M of circuits. Let S be a subset of E M .(1) The dual M * of M is a matroid on E M given by bases

show abstract

“…In this section, we review basic definitions and properties for matroids over hyperfields first introduced by Baker and Bowler in [5]. Let's first recall the definition of a hyperfield.…”

Section: Matroids Over Hyperfieldsmentioning

confidence: 99%

Section: Matroids Over Hyperfieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hopf algebras for matroids over hyperfields

2020

View full text Add to dashboard Cite

show abstract

“…We note that recently, in [BB16], Baker and Bowler implemented a notion of matroids over hyperfields and generalized various classes of matroids (valuated matroids, oriented matroids, and phased matroids) in a very elegant way.…”

Section: Quotient Constructionmentioning

confidence: 99%

“…This can be viewed as a partial generalization of the classical result that the category of Hopf algebras is equivalent to the opposite category of the category of affine group schemes. Moreover, in the recent paper [BB16], Baker and Bowler unified various generalizations of matroids (oriented, valuated, and phased) by means of hyperfields.…”

mentioning

confidence: 99%

Association schemes and hypergroups

Jun

2017

Communications in Algebra

View full text Add to dashboard Cite

Abstract. In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, §1.2 for details).

show abstract

Topological Posets and Tropical Phased Matroids

Alvarez,

Geoghegan

2024

Discrete Comput Geom

View full text Add to dashboard Cite

Matroids over hyperfields

Cited by 28 publications

References 21 publications

Hopf algebras for matroids over hyperfields

Hopf algebras for matroids over hyperfields

Association schemes and hypergroups

Topological Posets and Tropical Phased Matroids

Contact Info

Product

Resources

About