Matroids over partial hyperstructures. (English) Zbl 1404.05022
Summary: 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 C. Semple and G. Whittle [Adv. Appl. Math. 17, No. 2, 184–208 (1996; Zbl 0859.05035)]. 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 [A. W. M. Dress and W. Wenzel, Adv. Math. 91, No. 2, 158–208 (1992; Zbl 0757.05040)]. 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 A. W. M. Dress [ibid.. 59, 97–123 (1986; Zbl 0656.05025)] and A. Dress and W. Wenzel [ibid. 86, No. 1, 68–110 (1991; Zbl 0748.05040); ibid. 93, No. 2, 214–250 (1992; Zbl 0754.05027); Zbl 0757.05040 loc. cit.].
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
References:
| [1] | Anderson, L.; Delucchi, E., Foundations for a theory of complex matroids, Discrete Comput. Geom., 48, 4, 807-846 (2012) · Zbl 1256.05036 |
| [2] | Anderson, L., Vectors of matroids over tracts, J. Combin. Theory Ser. A, 161, 236-270 (2019) · Zbl 1400.05043 |
| [3] | Baker, M.; Bowler, N., Matroids over hyperfields (2017), 31 pages |
| [4] | Berkovich, V. G., Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, vol. 33 (1990), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0715.14013 |
| [5] | Bland, R. G.; Jensen, D. L., Weakly Oriented Matroids (1987), Cornell University School of OR/IE, Technical Report No. 732 |
| [6] | Bland, R. G.; Las Vergnas, M., Orientability of matroids, J. Combin. Theory Ser. B, 24, 1, 94-123 (1978) · Zbl 0374.05016 |
| [7] | Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G. M., Oriented Matroids, Encyclopedia of Mathematics and Its Applications, vol. 46 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0944.52006 |
| [8] | Basu, S.; Pollack, R.; Roy, M.-F., Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, vol. 10 (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1102.14041 |
| [9] | Connes, A.; Consani, C., From monoids to hyperstructures: in search of an absolute arithmetic, (Casimir Force, Casimir Operators and the Riemann Hypothesis (2010), Walter de Gruyter: Walter de Gruyter Berlin), 147-198 · Zbl 1234.14002 |
| [10] | Connes, A.; Consani, C., The hyperring of adèle classes, J. Number Theory, 131, 2, 159-194 (2011) · Zbl 1221.14002 |
| [11] | Delucchi, E., Modular elimination in matroids and oriented matroids, European J. Combin., 32, 3, 339-343 (2011) · Zbl 1290.05056 |
| [12] | Dress, A. W.M., Duality theory for finite and infinite matroids with coefficients, Adv. Math., 59, 2, 97-123 (1986) · Zbl 0656.05025 |
| [13] | Dress, A. W.M.; Wenzel, W., Grassmann-Plücker relations and matroids with coefficients, Adv. Math., 86, 1, 68-110 (1991) · Zbl 0748.05040 |
| [14] | Dress, A. W.M.; Wenzel, W., Valuated matroids, Adv. Math., 93, 2, 214-250 (1992) · Zbl 0754.05027 |
| [15] | Dress, A. W.M.; Wenzel, W., Perfect matroids, Adv. Math., 91, 2, 158-208 (1992) · Zbl 0757.05040 |
| [16] | Fink, A.; Moci, L., Matroids over a ring, J. Eur. Math. Soc., 018, 4, 681-731 (2016) · Zbl 1335.05031 |
| [17] | Frenk, B., Tropical Varieties, Maps, and Gossip (2013), 167 pages |
| [18] | Giansiracusa, J.; Giansiracusa, N., A Grassmann algebra for matroids, Manuscripta Math., 156, 1-2, 187 (2018) · Zbl 1384.05063 |
| [19] | Izhakian, Z.; Rowen, L., Supertropical algebra, Adv. Math., 225, 4, 2222-2286 (2010) · Zbl 1273.14132 |
| [20] | Izhakian, Z.; Rowen, L., Supertropical matrix algebra, Israel J. Math., 182, 383-424 (2011) · Zbl 1215.15018 |
| [21] | Jun, J., Algebraic geometry over hyperrings, Adv. Math., 323, 142-192 (2018) · Zbl 1420.14005 |
| [22] | Kleiman, S. L.; Laksov, D., Schubert calculus, Amer. Math. Monthly, 79, 1061-1082 (1972) · Zbl 0272.14016 |
| [23] | Marshall, M. A., Spaces of Orderings and Abstract Real Spectra, Lecture Notes in Mathematics, vol. 1636 (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0866.12001 |
| [24] | Marshall, M. A., Real reduced multirings and multifields, J. Pure Appl. Algebra, 205, 2, 452-468 (2006) · Zbl 1089.14009 |
| [25] | Massouros, Ch. G., Methods of constructing hyperfields, Int. J. Math. Math. Sci., 8, 4, 725-728 (1985) · Zbl 0587.12015 |
| [26] | Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161 (2015), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1321.14048 |
| [27] | Murota, K.; Tamura, A., On circuit valuation of matroids, Adv. in Appl. Math., 26, 3, 192-225 (2001) · Zbl 0979.05028 |
| [28] | Oxley, J. G., Matroid Theory, Oxford Science Publications (1992), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0784.05002 |
| [29] | Pendavingh, R. A.; van Zwam, S. H.M., Lifts of matroid representations over partial fields, J. Combin. Theory Ser. B, 100, 1, 36-67 (2010) · Zbl 1215.05024 |
| [30] | Pendavingh, R. A.; van Zwam, S. H.M., Skew partial fields, multilinear representations of matroids, and a matrix tree theorem, Adv. in Appl. Math., 50, 1, 201-227 (2013) · Zbl 1256.05047 |
| [31] | Semple, C.; Whittle, G., Partial fields and matroid representation, Adv. in Appl. Math., 17, 2, 184-208 (1996) · Zbl 0859.05035 |
| [32] | Tutte, W. T., A homotopy theorem for matroids. I, II, Trans. Amer. Math. Soc., 88, 144-174 (1958) · Zbl 0081.17301 |
| [33] | Viro, O. Y., Hyperfields for tropical geometry I. Hyperfields and dequantization (2010), 45 pages |
| [34] | Viro, O. Y., On basic concepts of tropical geometry, (Tr. Mat. Inst. SteklovaSovremennye Problemy Matematiki, vol. 273 (2011)), 271-303 · Zbl 1237.14074 |
| [35] | (White, N., Combinatorial Geometries. Combinatorial Geometries, Encyclopedia of Mathematics and Its Applications, vol. 29 (1987), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0626.00007 |
| [36] | Whittle, G., On matroids representable over \(GF(3)\) and other fields, Trans. Amer. Math. Soc., 349, 2, 579-603 (1997) · Zbl 0865.05029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
