Sets of positive integers closed under product and the number of decimal digits

A digital semigroup D is a subsemigroup of (N\{0},⋅) such that if d∈D then {x∈N\{0}|ℓ(x)=ℓ(d)}⊆D with ℓ(n) the number of digits of n written in decimal expansion. In this note, we compute the smallest digital semigroup containing a set of positive integers. For this, we establish a connection between the digital semigroups and a class of numerical semigroups called LD-semigroups.