© 2024 by the authors.Let (Formula presented.) be an integer polyhedral cone. An affine semigroup (Formula presented.) is a (Formula presented.) -semigroup if (Formula presented.). This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work, we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to (Formula presented.) -semigroups.