A numerical semigroup \(S\) of nonnegative integers contains \(0\) and it generates the set \(\mathbb{Z}\) of integers as a group. For a subset \(A\subseteq\mathbb{N}\), let \(\langle A\rangle=\{\sum_{i=1}^k a_ix_i\mid a_i\in A\), \(x_1,\dots,x_k\in\mathbb{N}\}\). Then \(\langle A\rangle\) is a numerical semigroup if and only if \(\gcd(A)=1\). It is well known that a numerical semigroup \(S\) has a unique minimal set of generators \(\{n_1,n_2,\dots,n_e\}\) and that \(\mathbb{N}\setminus S\) is finite. The greatest element \(g(S)\) of \(\mathbb{Z}\setminus S\) is called the Frobenius number of \(S\). Obtaining a general solution for \(g(S)\) in terms of \(n_1,n_2,\dots,n_e\) is an open problem for \(e \geq 3\). A numerical semigroup \(S\) is said to be pseudo-symmetric if \(g(S)\) is even and the only integer \(x\) such that \(x\in\mathbb{Z}\setminus S\) and \(g(S)-x\notin S\) is \(x=g(S)/2\).
The authors consider a numerical monoid \(S\) with a minimal set \(\{n_1,n_2,n_3\}\) of three generators. They provide an easy way to determine whether \(S\) is pseudo-symmetric, and they also give a short formula for \(g(S)\) in this case.