Submonoids of the formal 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.