  We study the free objects in the variety of semigroups and variety of monoids
generated by the monoid of all $n \times n$ upper triangular matrices over a
commutative semiring. We obtain explicit representations of these, as
multiplicative subsemigroups of quiver algebras over polynomial semirings. In
the $2 \times 2$ case this also yields a representation as a subsemigroup of a
semidirect product of commutative monoids. In particular, from the case where
$n=2$ and the semiring is the tropical semifield, we obtain a representation of
the free objects in the monoid and semigroup varieties generated by the
bicyclic monoid (or equivalently, by the free monogenic inverse monoid), 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.
