Numerical semigroups II: Pseudo-symmetric AA-semigroups

In this work we consider the general numerical AA-semigroup, i.e., semigroups consisting of all non-negative integer linear combinations of relatively prime positive integers of the form a,a+d,a+2d,…,a+kd,ca,a+d,a+2d,…,a+kd,c. We first prove that, in contrast to arbitrary numerical semigroups, there exists an upper bound for the type of AA-semigroups that only depends on the number of generators of the semigroup. We then present two characterizations of pseudo-symmetric AA-semigroups. The first one leads to a polynomial time algorithm to decide whether an AA-semigroup is pseudo-symmetric. The second one gives a method to construct pseudo-symmetric AA-semigroups and provides explicit families of pseudo-symmetric semigroups with arbitrarily large number of generators.