On row-factorization matrices and generic ideals

Let H be a numerical semigroup minimally generated by an almost arithmetic sequence. We give a complete description of the row-factorization (RF) matrices associated with the pseudo-Frobenius elements of H. RF-matrices have a close connection with the defining ideal of the semigroup ring associated to H. We use the information from RFmatrices to give a characterization of the generic nature of the defining ideal. When H has embedding dimension 3, we prove that under suitable assumptions, the defining ideal is minimally generated by RF-relations. We also consider the generic nature of the defining ideal of gluing of two numerical semigroups and conclude that such an ideal is never generic.