Factorization Invariants in half-Factorial Affine Semigroups

Let be the monoid generated by We introduce the homogeneous catenary degree of as the smallest N ∈ ℕ with the following property: for each and any two factorizations u, v of a, there exist factorizations u = w1,…,wt = v of a such that, for every k, d(wk,wk+1) ≤ N, where d is the usual distance between factorizations, and the length of wk, |wk|, is less than or equal to max{|u|, |v|}. We prove that the homogeneous catenary degree of improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the ω-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.