C-semigroups with its induced order

Let Cᵖ be an integer polyhedral cone. An affine semigroup S⊂ C is a C-semigroup if | C S|<+∞. 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 C-semigroups.