Free objects in triangular matrix varieties and quiver algebras over semirings

DOI10.1016/J.JALGEBRA.2021.08.030zbMATH Open1512.20193arXiv1904.06094MaRDI QIDQ6317095

Mark Kambites

Publication date: 12 April 2019

Abstract: We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $nimesn$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. In the $2imes2$ 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.

Mathematics Subject Classification ID

Semigroups of transformations, relations, partitions, etc. (20M20) Varieties and pseudovarieties of semigroups (20M07) Semifields (12K10) Semirings (16Y60)

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