Identities in unitriangular and gossip monoids

We establish a criterion for a semigroup identity to hold in the monoid of n × 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 non-trivial and idempotent, the generated variety is the variety 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 𝐉_𝐧-1 . As a consequence we obtain a complete solution to the finite basis problem for Lossy gossip monoids.