Perfect bricks of every size. (Q2447237)

Let \(S\) be a numerical semigroup, that is, a submonoid of the set of nonnegative integers with greatest common divisor equal to one (equivalently with finite complement in the set of nonnegative integers). A subset \(I\) of \(S\) is a relative ideal of \(S\) if \(I+S\subseteq I\) and there exists an integer \(x\) such that \(x+I\subseteq S\). Every numerical semigroup has a unique minimal generating system, and the same holds for every relative ideal of a numerical semigroup. Denote by \(\mu_S(I)\) the cardinality of the minimal generating system of the relative ideal \(I\) of the numerical semigroup \(S\). Given a relative ideal \(I\) of a numerical semigroup the dual of \(I\), \(S-I\), is the set of integers \(x\) such that \(x+I\subseteq S\). The dual of \(I\) is also an ideal of \(S\). It easily follows that \(\mu_S(I)\mu_S(S-I)\geq\mu_S(I+(S-I))\). When the equality holds, we say that \((S,I)\) is a perfect \(k\times n\) brick, where \(k=\mu_S(I)\) and \(n=\mu_S(S-I)\). The paper under review gives a parametrized family of perfect bricks for any positive integers \(n\) and \(k\). The semigroups obtained in this family are all symmetric (if \(f\) is the largest gap of \(S\), then \(S\) is symmetric if whenever \(x\) is an integer not is \(S\), \(f-x\in S\)). The authors propose some other families candidates to yield perfect bricks and with nonsymmetric associated semigroups.