Congruence compatible

Definition

Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, n_2, \ldots, n_p\}\), an element \(s \in S\), \(\nabla_s\) the graph associated to \(s\) and \(\tau \subseteq \mathbb{N}^p \times \mathbb{N}^p\). It is said that \(\tau\) is compatible with \(s \in S\) if either \(\nabla_s\) is connected or if \(R_1, R_2, \ldots, R_t\) are the connected components of \(\nabla_s\), then for every \(i \in \{1, 2, \ldots, t\}\) it can be chosen \(a_i \in R_i\) such that for every \(j \in \{1, 2, \ldots, t\}\) with \(i \ne j\), there exists \(i_1, i_2, \ldots, i_k \in \{1, 2, \ldots, t\}\) fulfilling

  • \(i_1 = i, i_k = j\),
  • for every \(m \in \{1, 2, \ldots, k-1\}\) either \((a_{i_m}, a_{i_{m+1}}) \in \tau\) or \((a_{i_{m+1}}, a_{i_m}) \in \tau\).

The elements \(s \in S\) such that \(\nabla_s\) is not connected are called Betti elements. It can be proven that \(\tau\) is compatible with \(s\) for all \(s \in S\) if, and only if, is a presentation of \(S\).

Examples

\(\circ\) Let \(S = \langle 5, 7, 11, 13 \rangle\) and \(s = 26\). The graph \(\nabla_{26}\) is as follows.

NSGraph 1 (1, 3, 0, 0) 2 (3, 0, 1, 0) 1--2 3 (0, 0, 0, 2)

The graph \(\nabla_{26}\) is not connected and it has two connected components, \(R_1 = \{(1,3,0,0), (3,0,1,0)\}\) and \(R_2 = \{(0,0,0,2)\}\). Let \(a_{1,1} = (1,3,0,0), a_{1,2} = (3,0,1,0)\) and \(a_{2,1} = (0,0,0,2)\), let us see what pairs needs a subset of \(\mathbb{N}^4 \times \mathbb{N}^4\) to be compatible with \(s = 26\). As \(\nabla_{26}\) is not connected, it has to satisfy the second condition.

  • For \(i = 1\), we need \(a \in R_1, b \in R_2\) such that \((a,b) \in \tau\) or \((b,a) \in \tau\)

  • For \(i = 2\) we have the same condition as before.

Then, \(\tau\) is compatible if has at least one of the following pairs: \((a_{1,1}, a_{2,1}), (a_{2,1}, a_{1,1}), (a_{1,2}, a_{2,1}), (a_{2,1}, a_{1,2})\).

Examples with GAP

Nowadays, there are no functions in package NumericalSgps related to congruence compatible.

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.