Join of affine semigroups

In this paper, we study the class of affine semigroup generated by integral vectors, whose components are in generalised arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that this affine semigroup exhibits some nice properties including Cohen-Macaulayness.