Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

Abstract

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n × n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2×2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of "chain structured tropical matrix semigroups"; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3 × 3 upper triangular matrix semigroups, and a new proof of the fact that this semigroup satisfies the same identities as the bicyclic monoid.

Identities of the Kauffman monoid K3

“…If, however, an infinite semigroup satisfies a nontrivial identity, its identity checking problem may constitute a challenge: Murskiǐ [1968] had constructed an infinite semigroup with undecidable identity checking problem. On the 'positive' side, we mention a recent result by Daviaud et al [2018] who have shown that checking identities in the famous bicyclic monoid B := a, b | ba = 1 can be done in polynomial time via rather a non-trivial algorithm based on linear programming.…”

Geometry and algorithms for upper triangular tropical matrix identities

“…Both of these operations are costly when the words are long. Our approach is based upon a geometric interpretation of the characterisation of UT n (T) identities given in [6], which involves more polynomial pairs but in fewer variables and monomials. A key observation is that the polynomials which appear in [6] are tropical polynomials with trivial coefficients, and thus they are equal if and only if their Newton polytopes are equal.…”

“…Our approach is based upon a geometric interpretation of the characterisation of UT n (T) identities given in [6], which involves more polynomial pairs but in fewer variables and monomials. A key observation is that the polynomials which appear in [6] are tropical polynomials with trivial coefficients, and thus they are equal if and only if their Newton polytopes are equal. This translates the problem to verifying equality of lattice polytopes, which can be efficiently computed and, more importantly, adds geometric insights that allow us to deduce structural information about these identities.…”

“…To the best of our knowledge, the only other algorithmic approaches to studying UT n (T) arise via the connection to the bicyclic monoid which satisfies the same identities as UT 2 (T) by [6]). Shleȋfer [17] enumerated all identities of the bicyclic monoid of length at most 13, by means of a computer program.…”

“…We now derive an alternative characterisation, Theorem 2.4. It builds on [6,Theorem 5.2], which utilises the well-known connection between tropical matrix multiplication and optimal paths in a weighted directed graph on n nodes [3,13]. The characterisation of Theorem 2.4 also translates w ∼ n v into functional equality of a system of tropical polynomials with trivial coefficients.…”