On atoms of the set of generalized numerical semigroups with fixed corner element

We study the atomic generalized numerical semigroups (GNSs), which naturally extend the concept of atomic numerical semigroups. We introduce the notion of corner special gap and we characterize the class of atomic GNS in terms of the cardinality of the set of corner special gaps and also in terms of a maximal property. Using this maximal property we present some properties concerning irreducibility of Frobenius GNSs. In particular, we provide sufficient conditions for certain Frobenius GNSs to be atom non-irreducible. Furthermore, we given necessary and sufficient conditions so that the maximal elements of a set of Frobenius GNSs with two fixed gaps to be all irreducible or not.