On Power Series over a Graded Monoid

Abstract

We consider power series over a graded monoid $M$ of finite type. We show first that, under certain conditions, the equivalence problem of power series over $M$ with coefficients in the semiring $\mathbb N$ of nonnegative integers can be reduced to the equivalence problem of power series over $\{x\}^*$ with coefficients in $\mathbb N$. This result is then applied to rational and recognizable power series over $M$ with coefficients in $\mathbb N$, and to rational power series over $\Sigma ^*$ with coefficients in the semiring $\mathbb {Q}_+$ of nonnegative rational numbers, where $\Sigma$ is an alphabet.

Partially supported by grant no. K 108448 from the National Foundation of Hungary for Scientific Research.

Partially supported by Austrian Science Fund (FWF): grant no. I1661-N25.