Numerical semigroups with Apéry sets of unique expression (Q1975919)

A numerical semigroup is a subset \(S\) of the set of nonnegative integers \(\mathbb{N}\) closed under addition, containing the identity element of \(\mathbb{N}\) and generating the set of integers as a group. For \(n\in S\setminus\{0\}\), the set \(\{s\in S\mid s-n\not\in S\}\) is called the Apéry set of \(n\). Let \(\{n_0<\cdots<n_p\}\) be the unique minimal system of generators of \(S\). Denote by \(F\) the free monoid on the indeterminates \(\{X_0<\cdots<X_p\}\) and let \(\phi\colon F\to S\), \(\phi(X_i)=n_i\) be the monoid epimorphism. Obviously, \(S\cong F/\ker\phi\). A presentation of \(S\) is a finite subset \(\rho\) of \(F\times F\) such that the congruence of \(\phi\) is the least congruence on \(F\) containing \(\rho\). The knowledge of a minimal presentation of a semigroup has applications in ring theory by using the concept of semigroup ring. In this paper, the author gives a minimal presentation for a numerical semigroup such that all the elements of its Apéry set with respect to \(n_0\) have a unique expression. Interesting examples of this type of semigroups are supplied.