Semiring systems arising from hyperrings. (English) Zbl 07808734
Summary: Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we show that the systems arising in this way, called hypersystems, are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems.
MSC:
| 06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |
| 14T10 | Foundations of tropical geometry and relations with algebra |
| 15A80 | Max-plus and related algebras |
| 16Y20 | Hyperrings |
| 16Y60 | Semirings |
| 20N20 | Hypergroups |
| 15A24 | Matrix equations and identities |
References:
| [1] | Akian, M.; Gaubert, S.; Guterman, A., Linear independence over tropical semirings and beyond, (Litvinov, G. L.; Sergeev, S. N., Proceedings of the International Conference on Tropical and Idempotent Mathematics. Proceedings of the International Conference on Tropical and Idempotent Mathematics, Contemporary Mathematics, vol. 495, (2009), American Mathematical Society), 1-38 · Zbl 1182.15002 |
| [2] | Akian, M.; Gaubert, S.; Guterman, A., Tropical Cramer determinants revisited, (Litvinov, G. L.; Sergeev, S. N., Proceedings of the International Conference on Tropical and Idempotent Mathematics. Proceedings of the International Conference on Tropical and Idempotent Mathematics, Contemporary Mathematics, vol. 616, (2014), American Mathematical Society), 1-45 · Zbl 1320.14074 |
| [3] | Akian, M.; Gaubert, S.; Rowen, L., Linear algebra over pairs, (2023) |
| [4] | Akian, Marianne; Gaubert, Stephane; Tavakolipour, Hanieh, Factorization of polynomials over the symmetrized tropical semiring and Descartes’ rule of sign over ordered valued fields, (2023) |
| [5] | Bruguières, A.; Burciu, Sebastian, On normal tensor functors and coset decompositions for fusion categories, Appl. Categ. Struct., 23, 4, 591-608, (2015) · Zbl 1333.18008 |
| [6] | Baker, M.; Bowler, N., Matroids over partial hyperstructures, Adv. Math., 343, 821-863, (2019) · Zbl 1404.05022 |
| [7] | Baker, M.; Lorscheid, O., Descartes’ rule of signs, Newton polygons, and polynomials over hyperfields, J. Algebra, 569, 416-441, (2021) · Zbl 1482.16078 |
| [8] | Baker, M.; Lorscheid, O., The moduli space of matroids, Adv. Math., 390, Article 107883 pp., (2021) · Zbl 1479.05045 |
| [9] | Benhissi, A.; Ribenboim, P., Ordered rings of generalized power series, (Ordered Algebraic Structures. Ordered Algebraic Structures, Gainesville, FL, 1991, (1993), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 99-109 · Zbl 0798.06022 |
| [10] | Connes, A.; Consani, C., The hyperring of adèle classes, J. Number Theory, 131, 2, 159-194, (2011) · Zbl 1221.14002 |
| [11] | Chapman, A.; Gatto, L.; Rowen, L., Clifford semialgebras, Rend. Circ. Mat. Palermo, 2, (2020) |
| [12] | Dress, A., Duality theory for finite and infinite matroids with coefficients, Adv. Math., 59, 97-123, (1986) · Zbl 0656.05025 |
| [13] | Dress, A.; Wenzel, W., Valuated matroids, Adv. Math., 93, 2, 214-250, (1992) · Zbl 0754.05027 |
| [14] | Gaubert, S., Théorie des systèmes linéaires dans les dioïdes, (1992), École des Mines de Paris, Phd thesis |
| [15] | Giansiracusa, J.; Jun, J.; Lorscheid, O., On the relation between hyperrings and fuzzy rings, Beitr. Algebra Geom., 58, 4, 735-764, , (, N, o, v, ., , 2, 0, 1, 7, ) · Zbl 1388.14018 |
| [16] | Golan, J., The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, vol. 54, (1992), Longman Sci & Tech. · Zbl 0780.16036 |
| [17] | Gatto, L.; Rowen, L., Grassmann semialgebras and the Cayley-Hamilton theorem, Proc. Am. Math. Soc. Ser. B, 7, 183-201, (2020) · Zbl 1468.16057 |
| [18] | Gunn, T., A Newton polygon rule for formally-real valued fields and multiplicities over the signed tropical hyperfield, (2021) |
| [19] | Gunn, T., Tropical extensions and baker-lorscheid multiplicities for idylls, (2022) |
| [20] | Henry, S., Symmetrization of monoïds as hypergroups, (2013) |
| [21] | Higman, G., Ordering by divisibility in abstract algebras, Proc. Lond. Math. Soc., 3, 2, 326-336, (1952) · Zbl 0047.03402 |
| [22] | Izhakian, Z.; Knebusch, M.; Rowen, L., Supertropical semirings and supervaluations, J. Pure Appl. Algebra, 215, 10, 2431-2463, (2011) · Zbl 1225.13009 |
| [23] | Izhakian, Z.; Knebusch, M.; Rowen, L., Categorical notions of layered tropical algebra and geometry, (Algebraic and Combinatorial Aspects of Tropical Geometry. Algebraic and Combinatorial Aspects of Tropical Geometry, Contemporary Mathematics, vol. 589, (2013), American Mathematical Society), 191-234 · Zbl 1284.14088 |
| [24] | Izhakian, Z.; Rowen, L., The tropical rank of a tropical matrix, Commun. Algebra, 37, 11, 3912-3927, (2009) · Zbl 1184.15003 |
| [25] | Izhakian, Z.; Rowen, L., Supertropical algebra, Adv. Math., 225, 4, 2222-2286, (2010) · Zbl 1273.14132 |
| [26] | Izhakian, Z.; Rhodes, J., New representations of matroids and generalizations, (2011) |
| [27] | Izhakian, Z.; Rowen, L., Supertropical matrix algebra ii: solving tropical equations, Isr. J. Math., 186, 1, 69-97, (2011) · Zbl 1277.15013 |
| [28] | Izhakian, Z., Tropical mathematics and representations of polyhedral objects, (2006), School of Mathematical Sciences and Computer Science, Tel Aviv University, Phd thesis |
| [29] | Jun, J.; Mincheva, K.; Rowen, L., Projective systemic modules, J. Pure Appl. Algebra, 224, 5, (2020) · Zbl 1447.16048 |
| [30] | Jun, J.; Mincheva, K.; Rowen, L., T-semiring pairs, Kybernetika, (2022) |
| [31] | Jun, J.; Rowen, L., Categories with negation morphisms, (Categorical, Homological and Combinatorial Methods in Algebra (AMS Special Session in Honor of S.K. Jain’s 80th Birthday), vol. 751, (2020), American Mathematical Society), 221-270 · Zbl 1462.08001 |
| [32] | Jun, J., Algebraic geometry over hyperrings, Adv. Math., 323, 142-192, (2018) · Zbl 1420.14005 |
| [33] | Krasner, M., A class of hyperrings and hyperfields, Int. J. Math. Math. Sci., 6, 2, 307-312, (1983) · Zbl 0516.16030 |
| [34] | Lorscheid, O., The geometry of blueprints. Part i: algebraic background and scheme theory, Adv. Math., 229, 3, (2012) · Zbl 1259.14001 |
| [35] | Lorscheid, O., A blueprinted view on \(\mathbb{F}_1\)-geometry, (Absolute Arithmetic and \(\mathbb{F}_1\)-Geometry, (2016), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 161-219, See also · Zbl 1351.11041 |
| [36] | MacLane, Saunders, Categories for the Working Mathematician, Grad. Texts Math., vol. 5, (1988), Springer-Verlag: Springer-Verlag New York etc. · Zbl 0705.18001 |
| [37] | Marshall, M., Real reduced multirings and multifields, J. Pure Appl. Algebra, 205, 452-468, (2006) · Zbl 1089.14009 |
| [38] | Mittas, J., Sur les hyperanneaux et les hypercorps, Math. Balk., 3, 368-382, (1973) · Zbl 0289.08001 |
| [39] | Massouros, C.; Massouros, G., On the borderline of fields and hyperfields, Mathematics, 11, 1289, (2023) |
| [40] | Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry, (2015), AMS · Zbl 1321.14048 |
| [41] | Plus, M., Linear systems in \((\max, +)\)-algebra, (Proceedings of the 29th Conference on Decision and Control, vol. 1. Proceedings of the 29th Conference on Decision and Control, vol. 1, Honolulu, , (, D, e, c, ., , 1, 9, 9, 0, )), 151-156, M. Plus is a collective name for M. Akian, G. Cohen, S. Gaubert, R. Nikoukhah, and J.-P. Quadrat |
| [42] | Richter-Gebert, J.; Sturmfels, B.; Theobald, T., First steps in tropical geometry, (Litvinov, G. L.; Maslov, V. P., Idempotent Mathematics and Mathematical Physics. Idempotent Mathematics and Mathematical Physics, Contemp. Math., vol. 377, (2005), AMS), 289-317 · Zbl 1093.14080 |
| [43] | Ribenboim, P., Rings of generalized power series. II. Units and zero-divisors, J. Algebra, 168, 1, 71-89, (1994) · Zbl 0806.13011 |
| [44] | Rowen, L. H., Algebras with a negation map, Eur. J. Math., (2021), Originally entitled Symmetries in tropical algebra |
| [45] | Reutenauer, C.; Straubing, H., Inversion of matrices over a commutative semiring, J. Algebra, 88, 2, 350-360, (1984) · Zbl 0563.15011 |
| [46] | Rhodes, J.; Silva, P. V., Boolean Representations of Simplicial Complexes and Matroids, Springer Monographs in Mathematics, (2015), Springer · Zbl 1343.05003 |
| [47] | Viro, O. Ya., Real plane algebraic curves: constructions with controlled topology, Algebra Anal., 1, 5, 1-73, (1989) |
| [48] | Viro, O. Ya., Dequantization of real algebraic geometry on logarithmic paper, (European Congress of Mathematics, (2001), Springer), 135-146 · Zbl 1024.14026 |
| [49] | Viro, O. Ya., Hyperfields for tropical geometry i. Hyperfields and dequantization, (2010) |
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