\(p\)-numerical semigroups with \(p\)-symmetric properties (Q6637167)

Let \(\mathbb{N}_0\) be the set of nonnegative integers. The paper under review deals with the concept of \(p\)-numerical semigroup, that is a generalization of the well known concept of \textit{numerical semigroup}. Let \(A=\{a_1,\ldots,a_k\}\) be a set of positive integers such that \(\gcd(A)=1\). For \(n\in \mathbb{N}_0\), denote by \(d(n;a_1,\ldots,a_k)\) the number of representations of \(n\) as \(a_1 x_1+\cdots a_k x_k=n\), with \(x_1,\ldots,x_n\in \mathbb{N}_0\). The set \(S_p(A)=\{n\in \mathbb{N}_0\mid d(n;a_1,\ldots,a_k)>p\}\) is called the \(p\)-numerical semigroup generated by \(A\). The concept of numerical semigroup corresponds to \(0\)-numerical semigroup (adding the zero element). Numerical semigroups can be characterized as the submnoids of \(\mathbb{N}_0\) having finite complement in it and many other properties of them are widely studied in several researches.\N\NIn the paper under review, the authors introduce the notions of \(p\)-symmetric and \(p\)-pseudo symmetric semigroups. These definitions generalize the concepts of symmetric and pseudo symmetric numerical semigroups, that are among the most studied classes of numerical semigroups. In fact, for these kind of numerical semigroups, many properties are known involving the main invariants like the genus, the Frobenius number, the Apéry sets, the type. The authors show how to define these invariants in the more general context of \(p\)-numerical semigroups. For instance, the \(p\)-Frobenius number is defined as \(g_p(S_p(A))=\max \{n\in \mathbb{N}_0\mid d(n;a_1,\ldots,a_k)\leq p\}\). For the definitions of \(p\)-symmetric and \(p\)-pseudo symmetric, a crucial role is played by the \(p\)-Apéry set and its minimum element, denoted by \(\ell_0(p)\), that is also the least element in the \(p\)-numerical semigroup. An interesting observation is that if \(p=0\) then \(\ell_0(p)=0\), but for \(p>0\) this element is not zero. A \(p\)-numerical semigroup \(S_p\) is called \(p\)-symmetric if for all \(x\in \mathbb{Z}\setminus S_p\), then \(\ell_0(p)+g_p(S_p)-x\in S_p\). It is called \(p\)-pseudo symmetric if for all \(x\in \mathbb{Z}\setminus S_p\) with \(x\neq (\ell_0(p)+g_p(S_p))/2\in \mathbb{Z}\), then \(\ell_0(p)+g_p(S_p)-x\in S_p\). Considering these definitions and the generalizations of the mentioned invariants, the authors find different characterizations of \(p\)-symmetric and \(p\)-pseudo symmetric semigroups, which extends the well known analogous properties in the context of numerical semigroups. Different examples are also provided together with these results.