Factorization invariants in half-factorial affine semigroups

Let $\mathbb{N}\mathcal{A}$ be the monoid generated by $$\mathcal{A}$ = {$\mathbf{a}$_1, …, $\mathbf{a}$_n} \subseteq $\mathbb{Z}$^d.$ We introduce the homogeneous catenary degree of $\mathbb{N}\mathcal{A}$ as the smallest $N \in \mathbb N$withthefollowingproperty:foreach$$\mathbf{a}$ \in $\mathbb{N}$ $\mathcal{A}$$andanytwofactorizations$$\mathbf{u}$, $\mathbf{v}$$of$$\mathbf{a}$$,thereexistsfactorizations$$\mathbf{u}$ = $\mathbf{w}$_1, …, $\mathbf{w}$_t = $\mathbf{v}$ $of$$\mathbf{a}$$suchthat,forevery$k, $\mathrm{d}$($\mathbf{w}$k, $\mathbf{w}${k+1}) \leq N,$where$$\mathrm{d}$$ is the usual distance between factorizations, and the length of $$\mathbf{w}$_k, |$\mathbf{w}$_k|,$islessthanorequalto$\max{|$\mathbf{u}$|, |$\mathbf{v}$|}.$Weprovethatthehomogeneouscatenarydegreeof$$\mathbb{N}$ $\mathcal{A}$$ 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 $\omega$-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.