Abstract
We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.
Similar content being viewed by others
Notes
Some authors use the term semimodule, to emphasise the non-invertibility of addition, but since no other kind of module exists over \(\mathbb {FT}\) we have preferred the more concise term.
References
Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds.) Handbook of Linear Algebra. Chapman and Hall, London (2006)
Butkovič, P.: Max-Linear Systems: Theory and Algorithms. Springer, Berlin (2010)
Butkovič, P., Schneider, H., Sergeev, S.: Generators, extremals and bases of max cones. Linear Algebra Appl. 421(2–3), 394–406 (2007)
Cameron, P.J.: Permutation groups (London Mathematical Society Student Texts vol 45). Cambridge University Press, Cambridge (1999)
Cameron, P.J., Giudici, M., Jones, G.A., Kantor, W.M., Klin, M.H., Marušič, D., Nowitz, L.A.: Transitive permutation groups without semiregular subgroups. J. Lond. Math. Soc. (2) 66(2), 325–333 (2002)
Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. In: Combinatorial and computational geometry, volume 52 of Math. Sci. Res. Inst. Publ., pp. 213–242. Cambridge University Press, Cambridge (2005)
Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004). (electronic)
Guillon, P., Izhakian, Z., Mairesse, J., Merlet, G.: The ultimate rank of tropical matrices. J. Algebra 437, 222–248 (2015)
Guterman, A., Shitov, Y.: Rank functions of tropical matrices. Linear Algebra Appl. 498, 326–348 (2016)
Hollings, C., Kambites, M.: Tropical matrix duality and Green’s \(\cal{D}\) relation. J. Lond. Math. Soc. 86, 520–538 (2012)
Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)
Izhakian, Z., Johnson, M., Kambites, M.: Pure dimension and projectivity of tropical polytopes. arXiv:1106.4525 [math.RA] (2011)
Johnson, M., Kambites, M.: Multiplicative structure of \(2 \times 2\) tropical matrices. Linear Algebra Appl. 435, 1612–1625 (2011)
Johnson, M., Kambites, M.: Green’s \(\cal{J}\)-order and the rank of tropical matrices. J. Pure Appl. Algebra 217, 280–292 (2013)
Johnson, M., Kambites, M.: Convexity of tropical polytopes. Linear Algebra Appl. 485, 531–544 (2015)
Merlet, G.: Semigroup of matrices acting on the max-plus projective space. Linear Algebra Appl. 432(8), 1923–1935 (2010)
Okniński, J.: Semigroups of Matrices, volume 6 of Series in Algebra. World Scientific Publishing Co., Inc., River Edge, NJ (1998)
Sergeev, S., Schneider, H., Butkovič, P.: On visualization scaling, subeigenvectors and Kleene stars in max algebra. Linear Algebra Appl. 431(12), 2395–2406 (2009)
Shitov, Y.: Tropical matrices and group representations. J. Algebra 370, 1–4 (2012)
Wagneur, E.: Moduloïds and pseudomodules I. Dimension theory. Discrete Math. 98(1), 57–73 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jean-Eric Pin.
Zur Izhakian: Research supported by the Alexander von Humboldt Foundation. Marianne Johnson: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites gratefully acknowledges the hospitality of Universität Bremen during a visit to Bremen.
Rights and permissions
About this article
Cite this article
Izhakian, Z., Johnson, M. & Kambites, M. Tropical matrix groups. Semigroup Forum 96, 178–196 (2018). https://doi.org/10.1007/s00233-017-9894-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-017-9894-1