Semigroup identities of supertropical matrices

Springer Science and Business Media LLC - Tập 105 - Trang 466-477 - 2022
Zur Izhakian1, Glenn Merlet2
1Department of Mathematics, Ariel University, Ariel, Israel
2Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

Tóm tắt

We prove that, for any n, the monoid of all $$n \times n$$ supertropical matrices extending tropical matrices satisfies nontrivial semigroup identities; in particular, such supertropical triangular matrices admit exactly the same identities satisfied by tropical triangular matrices. These semigroup identities are carried over to labeled-weighted digraphs with double arcs.

Tài liệu tham khảo

Butkovič, P.: Max-Linear Systems: Theory and Algorithms. Springer-Verlag London Ltd., London (2010)

Chen, Y., Hu, X., Luo, Y., Sapir, O.: The finite basis problem for the monoid of \(2\times 2\) upper triangular tropical matrices. Bull. Aust. Math. Soc. 94(1), 54–64 (2016)

Daviaud, L., Johnson, M., Kambites, M.: Identities in upper triangular tropical matrix semigroups and the bicyclic monoid. J. Algebra 501, 503–525 (2018)

Daviaud, L., Johnson, M.: The shortest identities for max-plus automata with two states. In: Larsen, K.G., Bodlaender, H.L., Raskin, J.-F. (eds.) MFCS 2017, Leibniz International Proceedings in Informatics (LIPIcs), vol. 83, pp. 48:1–48:13. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Wadern (2017)

Gathmann, A.: Tropical algebraic geometry. Jahresbericht der DMV 108, 3–32 (2006)

Guillon, P., Izhakian, Z., Mairesse, J., Merlet, G.: The ultimate rank of tropical matrices. J. Algebra 437, 222–248 (2015)

Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and its Applications. Princeton University Press, Princeton (2006)

Izhakian, Z.: Tropical arithmetic and matrix algebra. Commun. Algebra 37(4), 1445–1468 (2009)

Izhakian, Z.: Tropical algebraic sets, ideals and an algebraic Nullstellensatz. Int. J. Algebra Comput. 18, 1067–1098 (2008)

Izhakian, Z.: Semigroup identities in the monoid of triangular tropical matrices. Semigroup Forum 88(1), 145–161 (2014)

Izhakian, Z.: Erratum to: Semigroup identities in the monoid of triangular tropical matrices. Semigroup Forum 92(3), 733 (2016)

Izhakian, Z.: Tropical plactic algebra, the cloaktic monoid, and semigroup representations. J. Algebra 524, 290–366 (2019)

Izhakian, Z.: Semigroup identities of tropical matrix semigroups of maximal rank. Semigroup Forum 92(3), 712–732 (2016)

Izhakian, Z.: Commutative \(\nu \)-algebra and supertropical algebraic geometry, arXiv:1901.08032, (2019)

Izhakian, Z., Johnson, M., Kambites, M.: Tropical matrix groups. Semigroup Forum 96(1), 178–196 (2018)

Izhakian, Z., Margolis, S.: Semigroup identities in the monoid of 2-by-2 tropical matrices. Semigroup Forum 80(2), 191–218 (2010)

Izhakian, Z., Merlet, G.: Semigroup identities of tropical matrices through matrix ranks, arXiv:1806.11028, (2018)

Izhakian, Z., Niv, A., Rowen, L.: Supertropical \(\operatorname{SL}_n\). Linear Multilinear Algebra 66(7), 1461–1483 (2018)

Izhakian, Z., Rhodes, J.: New representations of matroids and generalizations, arXiv:1103.0503, (2011)

Izhakian, Z., Rhodes, J.: Boolean representations of matroids and lattices, arXiv:1108.1473, (2011)

Izhakian, Z., Rhodes, J., Steinberg, B.: Representation theory of finite semigroups over semirings. J. Algebra 336(1), 139–157 (2011)

Izhakian, Z., Rowen, L.: Supertropical algebra. Adv. Math. 225(4), 2222–2286 (2010)

Izhakian, Z., Rowen, L.: The tropical rank of a tropical matrix. Commun. Algebra 37(11), 3912–3927 (2009)

Izhakian, Z., Rowen, L.: Supertropical matrix algebra. Israel J. Math. 182(1), 383–424 (2011)

Izhakian, Z., Rowen, L.: Supertropical matrix algebra II: solving tropical equations. Israel J. Math. 186(1), 69–97 (2011)

Izhakian, Z., Rowen, L.: Supertropical matrix algebra III: powers of matrices and generalized eigenspaces. J. Algebra 341(1), 125–149 (2011)

Johnson, M., Tran, N.M.: Geometry and algorithms for upper triangular tropical matrix identities. J. Algebra 530, 470–507 (2019)

Merlet, G., Nowak, T., Sergeev, S.: Weak CSR expansions and transience bounds in max-plus algebra. Linear Algebra Appl. 461, 163–199 (2014)

Okninski, J.: Identities of the semigroup of upper triangular tropical matrices. Commun. Algebra 43(10), 4422–4426 (2015)

Shevrin, L.N., Volkov, M.V.: Identities of semigroups. Izv. VUZ. Matematika (Russian) 11, 3–47 (1985); Soviet Math. Iz. Vuz. 29 (English translation) 11, 1–64 (1985)

Shitov, Y.: A semigroup identity for tropical 3x3 matrices, arXiv:1406.2601, (2014)

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Springer Science and Business Media LLC

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