Multiplicative subsemigroups of the positive integers closed with respect to the number of digits

TL;DR

This paper introduces a method for constructing the smallest multiplicative subsemigroup of positive integers, closed under the number of digits in a fixed base, providing an alternative to previous approaches.

Contribution

It presents a new procedure for constructing digit-closed multiplicative subsemigroups of positive integers, expanding understanding of their structure.

Findings

Provides an explicit construction method

Characterizes the minimal subsemigroup for given sets

Offers an alternative approach to prior work

Abstract

In this paper we present an alternative approach to a problem dealt with by Rosales et al. In particular, once a base $b$ for the representation of the integers is fixed, we describe a procedure for constructing the smallest multiplicative subsemigroup of the positive integers closed with respect to the number of digits which contains a set of positive integers.