Digital numerical semigroups

Abstract

The number of digits in base ten system of a positive integer n, is denoted by l(n). A digital semigroup is a subsemigroup D of (N\{0}, ·) such that if d ∈ D, then {x ∈ N\{0} | l(x) = l(d)} ⊆ D. Let A ⊆ N\{0}. Denote by L(A) = {l(a) | a ∈ A}. We will say that a numerical semigroup S is a digital numerical semigroup if there is a digital semigroup D such that S = L(D) ∪ {0}. In this work we show that D = {S | S is a digital numerical semigroup} is a Frobenius variety, D(Frob=F) = {S ∈ D | F(S) = F} is a covariety and D(mult = m) = {S ∈ D | m(S) = m} is a Frobenius pseudo-variety. As a consequence we present some algorithms to compute D(Frob=F), D(mult = m) and D(gen = 1) = {S ∈ D | g(S) = 1}. If X ⊆ N\{0}, we denote by D[X] the smallest element of D containing X. If S = D[X], then we will say that X is a D-system of generators of S. We will prove that if S ∈ D, then S admits a unique minimal D-system of generators, denoted by Dmsg(S). The cardinality of Dmsg(S) is called the D-rank of S. We solve the Frobenius problem to elements of D with D-rank equal to 1. Moreover, we present an algorithmic procedure to calculate all the elements of D with fixed D-rank.