When the associated graded ring of a semigroup ring is Complete Intersection

Let (R,m) be the semigroup ring associated to a numerical semigroup S. In this paper we study the property of its associated graded ring to be Complete Intersection. In particular, we introduce and characterize β-rectangular and γ-rectangular Apéry sets, which will be the fundamental concepts of the paper and will provide, respectively, a sufficient condition and a characterization for to be Complete Intersection. Then we use these notions to give four equivalent conditions for in order to be Complete Intersection.