Matroids over partial hyperstructures

Abstract

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects which we call tracts; they generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general 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 results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. This is the case whenever vectors and covectors are orthogonal, and is closely related to the notion of "perfect fuzzy rings" from [DW92b]. For example, if F is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong matroid over F coincide. Our theory of matroids over tracts is closely related to but more general than "matroids over fuzzy rings" in the sense of Dress and Dress-Wenzel [Dre86, DW91, DW92a, DW92b].

Projective systemic modules

“…The other direction might not hold. The same reasoning holds for tracts of [3]. (iv) Another interesting example comes from valuation theory.…”

The Geometry of Gaussoids

“…It can be derived from the combinatorics of the the Grassmann-Plücker relations. This approach was initiated thirty years ago by Dress and Wenzel [8] and extended recently by Baker and Bowler [2]. The extent to which matroid theory and gaussoid theory can be further developed in parallel remains to be investigated.…”

“…In recent years, the theory of matroids has been linked tightly to the emerging field of tropical geometry [2,25]. Every matroid defines a tropical linear space, and conversely, every tropical linear space corresponds to a valuated matroid.…”

“…Let ν be the map P ∪ A → R that takes the following values, listed in the order above: (14,10,6,0,6,8,8,2,8,6,6,2,8,8,8,8,8,4,2,10,9,3,5,5,9,11,1,5,7,5,5,5,7,7,1,5,8,6,4,4).…”

Regular Parameter Elements and Regular Local Hyperrings

“…If both, the addition and the multiplication, are hyperoperations with the additive part being a canonical hypergroup, then we have superrings [5], which were introduced by Mittas in 1973 [6]. Until now, the most well known and studied type of hyperrings is the Krasner hyperring, that has a plentitude of applications in algebraic geometry [7,8], tropical geometry [9], theory of matroids [10], schemes theory [11], algebraic hypercurves [12,13], hypermomographies [14]. In addition, the theory of hypermodules was extensively investigated by Massouros [15].…”