### Upper triangular tropical matrix identities

Algebra and Geometry Seminar

5th December 2018, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

Matrices over the tropical semifield arise in models of discrete event systems, optimisation and scheduling problems. In contrast to the case of upper triangular matrices over a field of characteristic 0, it is known that the semigroup UT_n(T) of all upper triangular matrices over the tropical semifield satisfy non-trivial semigroup identities. For example, Izhakian and Margolis have shown that the identity:

ABBA AB ABBA = ABBA BA ABBA

holds for all A, B \in UT_2(T).

In this talk I will discuss the following questions, relating to some joint work with Laure Daviaud, Mark Kambites, and Ngoc Mai Tran:

* Given a pair of words over a fixed alphabet {A, B}, how to decide if these form a semigroup identity for UT_n(T)?

* Given a single word over a fixed alphabet {A, B}, how to decide if there exists another word with which it forms a semigroup identity for UT_n(T)?

In the case n=2, the answers to these questions can be readily expressed in terms of certain lattice polytope computations. (This case is of particular interest to semigroup theorists, because the identities satisfied by UT_2(T) turn out to be precisely those satisfied by the bicyclic monoid.)

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