We first prove that if an affine semigroup ring has a maximal regular sequence consisting of monomials, then the type is equal to the number of maximal elements of the Apéry set with respect to the set of exponents of the monomials, which form a maximal regular sequence. Further, we consider affine semigroups having finite complement in their rational polyhedral cone, called \(\mathscr {C}\) -semigroups, and prove that the notions of symmetric and almost symmetric \(\mathscr {C}\) -semigroups are independent of term orders. We further investigate the conductor and the Apéry set of a \(\mathscr {C}\) -semigroup with respect to a minimal extremal ray. Building upon this, we extend the notion of reduced type to \(\mathscr {C}\) -semigroups and study its extremal behavior. For all d and fixed \(e \ge 2d\) , we give a class of \(\mathscr {C}\) -semigroups of embedding dimension e such that both the type and the reduced type do not have any upper bound in terms of the embedding dimension. We further explore irreducible decompositions of a \(\mathscr {C}\) -semigroup and give a lower bound on the irreducible components in an irreducible decomposition. Consequently, we deduce that for each positive integer k, there exists a \(\mathscr {C}\) -semigroup S such that the number of irreducible components of S is at least k.
