Identities in unitriangular and gossip monoids

TL;DR

This paper characterizes identities in upper unitriangular matrix monoids over semirings, linking algebraic properties with combinatorial criteria, and applies these results to solve the finite basis problem for gossip monoids.

Contribution

It introduces a combinatorial criterion for identities in matrix monoids over semirings and applies it to solve the finite basis problem for gossip monoids.

Findings

Established a criterion for semigroup identities in matrix monoids.

Identified the generated variety as rm{J_{n-1}} for idempotent semirings.

Solved the finite basis problem for lossy gossip monoids.

Abstract

We establish a criterion for a semigroup identity to hold in the monoid of $n \times n$ upper unitriangular matrices with entries in a commutative semiring $S$. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of $S$. In the case where $S$ is idempotent, the generated variety is the variety $\mathbf{J_{n-1}}$, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid $R_n$ of all reflexive relations on an $n$ element set, or the Catalan monoid $C_n$. We propose $S$-matrix analogues of these latter two monoids in the case where $S$ is an idempotent semiring whose multiplicative identity element is the `top' element with respect to the natural partial order on $S$, and show that each generates $\mathbf{J_{n-1}}$. As a consequence we obtain a complete solution to the finite basis problem for lossy gossip monoids.