Submonoids of the formal power series

TL;DR

This paper characterizes when sets of formal power series with exponents in specific subsets of positive integers form monoids under composition, linking this property to submonoids and strong closure conditions.

Contribution

It provides a classification of subsets T of positive integers for which R[[x^T]] forms a monoid under composition, connecting this to submonoid and strong closure properties.

Findings

R[[x^T]] forms a monoid iff T is a submonoid of N.

Strongly closed monoids ensure R[[x^T]] is a monoid.

Extension of results to multivariable power series.

Abstract

Formal power series come up in several areas such as formal language theory , algebraic and enumerative combinatorics, semigroup theory, number theory etc. This paper focuses on the set x R[[x]] consisting of formal power series with zero constant term. This subset forms a monoid with the composition operation on series. We classify the sets T of strictly positive integers for which the set of formal power series, R[[x^T]]={all formal power series consisting of terms whose power is from T}, forms a monoid with composition as the operation. We prove that in order for R[[x^T]] to be a monoid, T itself has to be a submonoid of N. Unfortunately, this condition is not enough to guarantee the desired result. But if a monoid is strongly closed, then we get the desired result. We also consider an analogous problem for power series in several variables.