FREE OBJECTS IN TRIANGULAR MATRIX VARIETIES AND QUIVER ALGEBRAS OVER SEMIRINGS

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.