2019/04/12 by Kambites, Mark · 1 citation
#12K10 #16Y60 (secondary) #20M07 (primary) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1904.06094
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. In the 2 × 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.