Modular translations of numerical semigroups.
Let \(\mathbb N\) denote the set of nonnegative integers. A numerical semigroup is a submonoid of \(\mathbb N\) with finite complement in \(\mathbb N\). Let \(S\) be a numerical semigroup and let \(m\in S\setminus\{0\}\). For \(a\in\mathbb N\), the set \(\mathrm T(S,a,m)=\{s+(as\bmod m)\mid s\in S\}\) is a numerical semigroup if and only if \(a+1\) and \(m\) are coprime. One says that a numerical semigroup \(S'\) is a modular translation of a numerical semigroup \(S\) if \(S'=\mathrm T(S,a,b)\) for some \(m\in S\setminus\{0\}\) and \(a\in\mathbb N\) (with \(\gcd(a+1,m)=1\)). The aim of this manuscript is to describe the main invariants of \(S'\) in terms of those of \(S\), and this is performed with the help of the Apéry sets. The Apéry set of \(m\) in \(S\) is \(\mathrm{Ap}(S,m)=\{s\in S\mid s-m\not\in S\}\). From this set, by using Selmer's formulas, one can compute the maximum integer not belonging to \(S\) (known as its Frobenius number, denoted by \(\mathrm F(S)\)) and the genus of \(S\) (the number of nonnegative integers not in \(S\)). It is well known that \(\mathrm{Ap}(S,m)=\{0=w(0),w(1),\dots,w(m-1)\}\), where \(w(i)\) is the least integer in \(S\) congruent with \(i\) modulo \(m\). The authors show that \(\mathrm{Ap}(\mathrm T(S,a,m),m)=\{w(i)+(ai\bmod m)\mid i\in\{0,\dots,m-1\}\}\). This, together with Selmer's formulas, is used to determine (and give bounds) for the Frobenius number of \(\mathrm T(S,a,m)\) in terms of the Frobenius number of \(S\). In the same way, a formula for the genus of \(\mathrm T(S,a,m)\) is provided, and with this, the authors are able to prove that if \(\gcd(a,m)=d\) and \(a\mathrm F(S)\bmod m=m-d\) (in this case \(\mathrm F(\mathrm T(S,a,m))=\mathrm F(S)+m-d\)), then \(S\) is irreducible if and only if \(\mathrm T(S,a,b)\) is irreducible. Recall that a numerical semigroup is irreducible if it is not the intersection of two numerical semigroups properly containing it. The numerical semigroup \(S\) has a unique minimal system of generators, and it is precisely \(\mathrm{Minimals}_{\leq_S}((\mathrm{Ap}(S,m)\setminus\{0\})\cup\{m\})\), where for \(a,b\in\mathbb Z\), \(a\leq_S b\) if \(b-a\in S\). The cardinality of the minimal generating system of \(S\) is the embedding dimension of \(S\). It is well known that the set of maximal elements in the Apéry sets is in one-to-one correspondence with \(\mathrm{PF}(S)=\{z\in\mathbb Z\setminus S\mid z+(S\setminus \{0\})\subseteq S\}\). In fact, \(\mathrm{Maximals}_{\leq_S}(\mathrm{Ap}(S,m))=m+\mathrm{PF}(H)\). The cardinality of \(\mathrm{PF}(S)\) is the type of \(S\). The authors prove that the embedding dimension of \(S\) is less than or equal to that of \(\mathrm T(S,a,m)\), and the same holds for the type. The paper contains many examples, and some contain special families where the bounds are attained or showing that they might be strict.
scientific article; zbMATH DE number 6147887
scientific article