2018/04/30 by Johnson, Marianne, Fenner, Peter · 2 citations
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1804.11100
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 idempotent, the generated variety is the variety \mathbfJn-1, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid Rn of all reflexive relations on an n element set, or the Catalan monoid Cn. 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 \mathbfJn-1. As a consequence we obtain a complete solution to the finite basis problem for lossy gossip monoids.