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Orientable arithmetic matroids. (English) Zbl 1437.05041
Summary: The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of “oriented arithmetic matroid” and prove some properties like the “uniqueness of orientation”.

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.)
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[1] Baker, M.; Bowler, N., Matroids over partial hyperstructures, Adv. Math., 343, 821-863 (2019) · Zbl 1404.05022
[2] 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
[3] Brändén, P.; Moci, L., The multivariate arithmetic Tutte polynomial, Trans. Amer. Math. Soc., 366, 10, 5523-5540 (2014) · Zbl 1300.05133
[4] Callegaro, F.; D’Adderio, M.; Delucchi, E.; Migliorini, L.; Roberto, P., Orlik-Solomon type presentations for the cohomology algebra of toric arrangements, Trans. Amer. Math. Soc., 32 (2019)
[5] D’Adderio, M.; Moci, L., Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math., 232, 335-367 (2013) · Zbl 1256.05039
[6] De Concini, C.; Procesi, C., On the geometry of toric arrangements, Transform. Groups, 10, 3-4, 387-422 (2005) · Zbl 1099.14043
[7] De Concini, C.; Procesi, C., Topics in hyperplane arrangements, polytopes and box-splines, Universitext (2011), Springer: Springer New York · Zbl 1217.14001
[8] De Concini, C.; Procesi, C.; Vergne, M., Vector partition functions and index of transversally elliptic operators, Transform. Groups, 15, 4, 775-811 (2010) · Zbl 1223.58014
[9] Fink, A.; Moci, L., Matroids over a ring, J. Eur. Math. Soc. (JEMS), 18, 4, 681-731 (2016) · Zbl 1335.05031
[10] Folkman, J.; Lawrence, J., Oriented matroids, J. Combin. Theory Ser. B, 25, 2, 199-236 (1978) · Zbl 0325.05019
[11] Jaeger, F., Tutte polynomials and link polynomials, Proc. Amer. Math. Soc., 103, 2, 647-654 (1988) · Zbl 0665.57006
[12] Kauffman, L. H., A Tutte polynomial for signed graphs, Combinatorics and complexity (Chicago, IL, 1987). Combinatorics and complexity (Chicago, IL, 1987), Discrete Appl. Math., 25, 1-2, 105-127 (1989) · Zbl 0698.05026
[13] Lawrence, J., Oriented matroids and multiply ordered sets, Linear Algebra Appl., 48, 1-12 (1982) · Zbl 0506.05019
[14] Lenz, M., Representations of weakly multiplicative arithmetic matroids are unique, Ann. Comb., 23, 2, 335-346 (2019) · Zbl 07077824
[15] Lenz, M., On powers of Plücker coordinates and representability of arithmetic matroids, Adv. Appl. Math., 112, 101911, 44 (2020) · Zbl 1435.05043
[16] Maurer, S. B., Matroid basis graphs. I, J. Combin. Theory Ser. B, 14, 216-240 (1973) · Zbl 0244.05015
[17] Maurer, S. B., Matroid basis graphs. II, J. Combin. Theory Ser. B, 15, 121-145 (1973) · Zbl 0247.05018
[18] Moci, L., A Tutte polynomial for toric arrangements, Trans. Amer. Math. Soc., 364, 2, 1067-1088 (2012) · Zbl 1235.52038
[19] Oxley, J., Matroid theory, Oxford Graduate Texts in Mathematics (2011), Oxford University Press: Oxford University Press Oxford · Zbl 1254.05002
[20] Pagaria, R., Combinatorics of toric arrangements, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30, 2, 317-349 (2019) · Zbl 1423.52040
[21] Thistlethwaite, M. B., A spanning tree expansion of the Jones polynomial, Topology, 26, 3, 297-309 (1987) · Zbl 0622.57003
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