Semigroup identities in the monoid of triangular tropical matrices
Tóm tắt
We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities. The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities.
Tài liệu tham khảo
Adjan, S.I.: Defining relations and algorithmic problems for groups and semigroups. In: Proceeding of the Steklov Institute of Mathematics, vol. 85. AMS, Providence (1967)
Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds.) Handbook of Linear Algebra. Chapman & Hall, London (2006)
Butkovič, P.: Max-Linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010)
Gaubert, S., Katz, R.D.: Reachability problems for products of matrices in semirings. Int. J. Algebra Comput. 16(3), 603–627 (2006)
Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHÉS 53, 53–73 (1985)
Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry vol. 35. Birkhauser, Basel (2007). Oberwolfach seminars
Izhakian, Z., Margolis, S.W.: Semigroup identities in the monoid of 2-by-2 tropical matrices. Semigroup Forum 80(2), 191–218 (2010)
Pin, J.-E.: Tropical Semirings, vol. 11, pp. 50–69. Cambridge University Press, Cambridge (1998)
Richter-Gebert, J., Sturmfels, B., Theobald, T.: First Steps in Tropical Geometry. Idempotent Mathematics and Mathematical Physics. Contemp. Math., vol. 377, pp. 289–317. AMS, Providence (2005)
Shevrin, L.N., Volkov, M.V.: Identities of semigroups. Sov. Math. Izv. VUZ 29(11), 1–64 (1985). Izv. Vuzov. Math. 29(11), 3–47 (1985)
Shneerson, L.: Identities in finitely generated semigroups of polynomial growth. J. Algebra 154(1), 67–85 (1993)
Simon, I.: Recognizable sets with multiplicities in the tropical semiring. In: Chytil, M., Janiga, L., Koubek, V. (eds.) MFCS 88. Lecture Notes in Computer Science, vol. 324, pp. 107–120. Springer, Berlin (1988)