Unboundedness of the first Betti number and the last Betti number of numerical semigroups generated by concatenation (Q6572386)

Let \(\mathbb{N}\) denote the set of non-negative integers, and let \(\mathbb{Z}\) denote the set of integers. A numerical semigroup is a submonoid of \((\mathbb{N},+)\) with finite complement in \(\mathbb{N}\). Associated to a numerical semigroup \(S\) one can define the following order relation on \(\mathbb{Z}\): \(a\le_S b\) if \(b-a\in S\).\N\NThe set of minimal elements of \(S^*=S\setminus\{0\}\) with respect to \(\le_S\) is the minimal generating set of \(S\). This set is finite, since there is at most one element in this set in each congruence class modulo \(\operatorname{m}(S)=\min(S^*)\) (the multiplicity of \(S\)). The cardinality of the minimal generating set of \(S\) is called the embedding dimension of \(S\). If \(\{n_1,\dots,n_e\}\) is the minimal generating set of \(S\), then every \(s\in S\) can be expressed as \(a_1n_1+\dots+a_en_e\), with \(a_1,\dots,a_e\in \mathbb{N}\).\N\NGiven a field \(K\) and \(t\) an indeterminate, we define the semigroup ring associated to \(S\) as \(K[S]=K[t^{n_1},\dots,t^{n_e}]=\bigoplus_{s\in S} Kt^s\). Let \(x_1,\dots,x_e\) be indeterminates and let \(\varphi: K[x_1,\dots,x_e]\to K[S]\) be the ring homomorphism induced by \(x_i\mapsto t^{n_i}\), for all \(i\in\{1,\dots,e\}\). The kernel of this congruence is a binomial ideal known as the defining ideal of \(S\).\N\NThe pseudo-Frobenius numbers of \(S\) are the maximal elements in \(\mathbb{Z}\setminus S\) with respect to \(\le_S\). The (Cohen-Macaulay) type of \(S\) is the number of pseudo-Frobenius numbers of \(S\). The set of pseudo-Frobenius numbers can be computed from the set of maximal elements (with respect to the order induced by the semigroup) in the Apéry set of \(s\) in \(S\), for any \(s\in S^*\). Recall that the Apéry set of \(s\) in \(S\) is defined as \(\operatorname{Ap}(S,s)=\{ n\in S : n-s\not\in S \}\).\N\NThe authors give a couple of families, one of embedding dimension four and another of embedding dimension five, for which the number of minimal generators of their corresponding defining ideals and the type is not bounded.\N\NThe first family is the family of numerical semigroups \(\mathfrak{S}_{(n,4)}\) with minimal generating set \(\{n^2+2n,n^2+2n+1, n^2+3n+1,n^2+3n+2\}\), with \(n\) an integer larger than four. For this family, in [\textit{R. Mehta} et al., J. Algebra Appl. 20, No. 9, Article ID 2150162, 26 p. (2021; Zbl 1509.20097)] it was proven that the cardinality of the minimal generating set of the defining ideal of this semigroup is \(2(n+1)\). In the manuscript under review, the authors give a description of \(\operatorname{Ap}(\mathfrak{S}_{(n,4)},n^2+2n)\), and thus they are able to compute the pseudo-Frobenius numbers and type of \(\mathfrak{S}_{(n,4)}\), which equals \(2n\).\N\NThe second family is the family of numerical semigroups \(\mathfrak{S}_{(n,5)}\) of numerical semigroups with minimal generating set \(\{n^2+3n+1,n^2+3n+1,n^2+3n+3,n^2+4n+3, n^2+4n+4\}\). A description of \(\operatorname{Ap}(\mathfrak{S}_{(n,5)},n^2+3n+1)\) is given, and from this description the set of pseudo-Frobenius numbers and type of \(\mathfrak{S}_{(n,5)}\) is calculated. The type of \(\mathfrak{S}_{(n,5)}\) is \((3n+6)/2\).\N\NIn the last section, the authors describe a minimal generating set of the defining ideal of \(\mathfrak{S}_{(n,5)}\).