Let \(S\) be a numerical semigroup, that is, a submonoid of the set of nonnegative integers, \(\mathbb N\), with respect to addition, and such that \(\mathbb N\setminus S\) has finitely many elements. The cardinality of \(\mathbb N\setminus S\) is the genus of \(S\), and the maximum of \(\mathbb Z\setminus S\) the Frobenius number of \(S\).
For a positive integer \(d\), the set \(S/d=\{x\in\mathbb N\mid dx\in S\}\) is again a numerical semigroup, called the quotient of \(S\) by \(d\). Let \(S\) and \(T\) be numerical semigroups. If \(S=T/d\), then we also say that \(T\) is a \(d\)-fold of \(S\). It may happen that several semigroups \(T\) verify that \(T/d=S\). It is thus a natural problem to study such family of numerical semigroups and their invariants, and see how they are related to those of \(S\). The author proves that the minimal possible genus on all \(d\)-folds of \(S\) is \(g+\lceil(d-1)f/d\rceil\), where \(g\) is the genus of \(S\) and \(f\) is its Frobenius number.
A numerical semigroup \(S\) with genus \(g\) and Frobenius number \(f\) is symmetric if \(g=(f+1)/2\). This is equivalent to the fact that for every integer \(z\), \(z\not\in S\) if and only if \(f-z\in S\). This concept is generalized for a positive integer \(d\) as follows: \(S\) is \(d\)-symmetric if for any \(z\in d\mathbb Z\), either \(z\in S\) or \(f-z\in S\). The author proves that if \(S\) is \(d\)-symmetric, then the Frobenius number of \(S/d\) is \((f-x)/d\), where \(x=\min\{x\in S\mid f\equiv x\bmod d\}\). Frobenius numbers of quotients of several families and particular \(d\)’s are studied in Section 4.
If \(S\) is a numerical semigroup, the relation \(\leq_S\) defined as \(a\leq_Sb\) if \(b-a\in S\), \(a,b\in\mathbb Z\), is an order relation. The set of maximal elements of \(\mathbb Z\setminus S\) with respect to this relation is the set of {pseudo-Frobenius} numbers of \(S\), and its cardinality is the type of \(S\). A numerical semigroup with genus \(g\), Frobenius number \(f\) and type \(t\) is almost symmetric if \(g=(f+t)/2\). In particular, symmetric (and pseudo-symmetric) numerical semigroups are pseudo-symmetric. For \(S\) almost symmetric, the author characterizes the set of \(d\)-folds of \(S\) with minimal genus. He also gives a formula for the minimum genus of the symmetric doubles of a numerical semigroup. This is performed by using the numerical duplication of \(S\). Take \(b\) an odd element of \(S\) and \(E\) a relative ideal of \(S\), that is, \(E\subseteq S\) and \(E+S\subseteq E\). Then \(S\bowtie^bE=2S\cup(2E+b)\) is a numerical semigroup whenever \(E+E+b\subseteq S\), called the numerical duplication of \(S\) with respect to \(E\) and \(d\). Every symmetric numerical semigroup \(T\) with \(T/2=S\) is of the form \(S\bowtie^bE\). – The paper provides many explanatory examples.