There is no polynomial formula for the catenary and the tame degree of finitely generated monoids

TL;DR

This paper proves that for a broad class of finitely generated monoids, there is no polynomial formula that can determine their catenary and tame degrees, highlighting limitations in current algebraic methods.

Contribution

It establishes the non-existence of polynomial formulas for catenary and tame degrees in general finitely generated monoids, extending previous results limited to special cases.

Findings

No polynomial formula exists for the catenary degree in broad classes of monoids.

No polynomial formula exists for the tame degree in broad classes of monoids.

Results apply to a sufficiently large class of finitely generated monoids.

Abstract

In the last two decades there has been a wealth of results determining the precise value of the catenary degree and the tame degree. Mostly, however, only for very special classes of monoids and domains. In the present work we now show that there is no polynomial formula, neither for the catenary nor for the tame degree, which is valid for a sufficiently large class of finitely generated monoids.