We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all n×n upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. A case of particular interest is where n = 2 and the semiring is the tropical semifield, where the variety coincides with that generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid); here we are obtain faithful representations of the free objects in this variety inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.
|Journal||Journal of Algebra|
|Publication status||Accepted/In press - 31 Aug 2021|