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We study the so-called atomic GNS, which naturally extends the concept ofatomic numerical semigroup. We introduce the notion of corner special gap andwe characterize the class of atomic GNS in terms of the cardinality of the setof corner special gaps and also in terms of a maximal property. Using thismaximal property we present some properties concerning irreducibility ofFrobenius GNSs. In particular, we provide sufficient conditions for certainFrobenius GNSs to be an atom non-irreducible (ANI). Furthermore, we givennecessary and sufficient conditions so that the maximal elements of a set ofFrobenius GNSs with two fixes gaps to be all irreducible or not.

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