Digital Numerical Semigroups

Abstract

The number of digits in base ten system of a positive integer $n$, is denoted by $\mathrm{l}(n).$
A digital semigroup is a subsemigroup $\D$ of $(\N\backslash \{0\}, \cdot)$ such that if $d\in \D,$ then $\{x\in \N\backslash \{0\}\mid \mathrm{l}(x)=\l(d)\}\subseteq \D.$
Let $A\subseteq \N\backslash \{0\}.$ Denote by $ \L(A)=\{\mathrm{l}(a)\mid a \in A\}.$ We will say that a numerical semigroup $S$ is a digital numerical semigroup if there is a digital semigroup $\D$ such that $S=\L(\D)\cup \{0\}.$
In this work we show that $\CD=\{S\mid S \mbox{ is a digital numerical semigroup}\}$ is a Frobenius variety, $\CD(\mbox{Frob=}F)=\{S\in \CD\mid \F(S)=F\}$ is a covariety and $\CD(\mbox{mult}=m)=\{S\in \CD\mid \m(S)=m\}$ is a Frobenius pseudo-variety. As a consequence we present some algorithms to compute $\CD(\mbox{Frob=}F)$, $\CD(\mbox{mult}=m)$ and $\CD(\mbox{gen}=g)=\{S\in \CD\mid \g(S)=g\}.$

If $X\subseteq \N\backslash \{0\},$ we denote by $\CD[X]$ the smallest element of $\CD$ containing $X$. If $S=\CD[X],$ then we will say that $X$ is a $\CD$-system of generators of $S$. We will prove that if $S \in \CD,$ then $S$ admits a unique minimal $\CD$-system of generators, denoted by $\CD\msg(S).$ The cardinality of $ \CD\msg(S)$ is called the $\CD$-rank of $S.$ We solve the Frobenius problem to elements of $\CD$ with $\CD$-$\rank$ equal to $1.$ Moreover, we present an algorithmic procedure to calculate all the elements of
$\CD$ with fixed $\CD$-rank.