Free objects in triangular matrix varieties and quiver algebras over semirings

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