Identities in Upper Triangular Tropical Matrix Semigroups and the Bicyclic Monoid

TL;DR

This paper characterizes identities in upper triangular tropical matrix semigroups using tropical polynomials, provides an algorithm for identity verification, and shows these identities coincide with those of the bicyclic monoid, extending to broader classes.

Contribution

It establishes necessary and sufficient conditions for identities in tropical matrix semigroups, introduces a polynomial-time algorithm for checking them, and links these identities to the bicyclic monoid.

Findings

Identities in 2x2 upper triangular tropical matrices match those of the bicyclic monoid.

An efficient algorithm for verifying semigroup identities was developed.

The results extend to broader classes of chain structured tropical matrix semigroups.

Abstract

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of $n\times n$ 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 $2\times 2$ 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 $3\times 3$ upper triangular matrix semigroups, and a new proof of the fact that this semigroup satisfies the same identities as the bicyclic monoid.