Factorization sets and half-factorial sets in integral domains (Q1904072)

We quote the authors' words: ``Let \({R}\) be an atomic integral domain. Suppose that \(H\) is an nonempty subset of irreducible elements of \({R}\), \(u\) is a unit of \({R}\), and \(a_1,\dots,a_n\), \(\beta_1,\dots,\beta_m\) are irreducible elements of \({R}\) such that (1) \(\alpha_1\dots \alpha_n=u\cdot\beta_1\dots \beta_m\). Set \(H_\alpha= \{i\mid\alpha_i\in H\}\) and \(H_\beta=\{i\mid\beta_i\in H\}\). \(H\) is a factorization set (F-set) of \({R}\) if for any equality involving irreducibles of the form (1), \(|H_\alpha|\neq 0\) implies that \(|H_\beta|\neq 0\). \(H\) is a half-factorial set (HF-set) if any equality of the form (1) implies that \(|H_\alpha|=|H_\beta|\). In this paper, we explore in detail the structure of the F-sets and HF-sets of an atomic integral domain \(R\)''.