Matroids over hyperfields

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. 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-Plucker 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.

[1]  Marc Krasner,et al.  A class of hyperrings and hyperfields , 1983 .

[2]  Jaiung Jun.  Algebraic Geometry Over Hyperrings , 2015, 1512.04837.

[3]  Emanuele Delucchi,et al.  Foundations for a Theory of Complex Matroids , 2012, Discrete & Computational Geometry.

[4]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[5]  W. T. Tutte.  A homotopy theorem for matroids. II , 1958 .

[6]  O. Viro.  On basic concepts of tropical geometry , 2011 .

[7]  Walter Wenzel,et al.  Grassmann-Plücker relations and matroids with coefficients , 1991 .

[8]  Emanuele Delucchi,et al.  Modular elimination in matroids and oriented matroids , 2010, Eur. J. Comb..

[9]  Oleg Viro.  Hyperfields for Tropical Geometry I. Hyperfields and dequantization , 2010 .

[10]  V. Berkovich.  Spectral Theory and Analytic Geometry over Non-Archimedean Fields , 1990 .

[11]  Stefan H. M. van Zwam,et al.  Skew partial fields, multilinear representations of matroids, and a matrix tree theorem , 2013, Adv. Appl. Math..

[12]  Jeffrey Giansiracusa,et al.  A Grassmann algebra for matroids , 2015, 1510.04584.

[13]  Alain Connes,et al.  The hyperring of adèle classes , 2011 .

[14]  Michel Las Vergnas,et al.  Orientability of matroids , 1978, J. Comb. Theory B.

[15]  Christos G. Massouros,et al.  Methods of constructing hyperfields , 1985 .

[16]  Supertropical matrix algebra II: Solving tropical equations , 2009, 0902.2159.

[17]  Bj Frenk,et al.  Tropical varieties, maps and gossip , 2008 .

[18]  A. Connes,et al.  From monoids to hyperstructures: in search of an absolute arithmetic , 2010, 1006.4810.

[19]  Rudi Pendavingh,et al.  Lifts of matroid representations over partial fields , 2008, J. Comb. Theory, Ser. B.

[20]  Kazuo Murota,et al.  On Circuit Valuation of Matroids , 2001, Adv. Appl. Math..

[21]  Alex Fink,et al.  Matroids Over a Ring , 2012 .

[22]  Jaiung Jun.  Valuations of semirings , 2015, Journal of Pure and Applied Algebra.

[24]  Oliver Lorscheid.  Scheme theoretic tropicalization , 2015 .

[25]  M. Marshall.  Real reduced multirings and multifields , 2006 .

[26]  Andreas W. M. Dress,et al.  Duality theory for finite and infinite matroids with coefficients , 1986 .