Characterizing affine $${\mathcal {C}}$$-semigroups

Abstract Let $${\mathcal {C}} \subset {\mathbb {N}}^p$$ C ⊂ N p be a finitely generated integer cone and $$S\subset {\mathcal {C}}$$ S an affine semigroup such that the real cones by S are equal. The is called -semigroup if {C}}\setminus S$$ \ finite set. In this paper, we characterize -semigroups from their minimal generating sets, give algorithm to check compute its set of gaps. We also study embedding dimension obtaining lower bound for it, introduce some families whose reaches our bound. last section, present method obtain decomposition into irreducible -semigroups.