FACTORIZATION INVARIANTS IN HALF-FACTORIAL AFFINE SEMIGROUPS

Abstract

Let [Formula: see text] be the monoid generated by [Formula: see text] We introduce the homogeneous catenary degree of [Formula: see text] as the smallest N ∈ ℕ with the following property: for each [Formula: see text] 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 [Formula: see text] 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.