Half-factorial domains and half-factorial subsets of Abelian groups (Q2705812)

An integral domain (resp., commutative, cancellative monoid) \(D\) is half-factorial if each nonzero nonunit of \(D\) is a product of irreducible elements and any two factorizations of a nonzero nonunit of \(D\) into the product of irreducible elements have the same length. A subset \(G_0\) of an Abelian group \(G\) is said to be half-factorial if the block monoid \({\mathcal B}(G_0)\) over \(G_0\) is half-factorial. Let \(D\) be a Krull domain (resp., Krull monoid) with divisor class group \(G\) and let \(G_0\subset G\) be the set of classes containing prime divisors. Then \(D\) is half-factorial if and only if \(G_0\) is half-factorial. It is not known if every Abelian group \(G\) has a half-factorial generating subset. This paper studies the structure and size of half-factorial subsets of an Abelian group \(G\). Let \(\mu(G)=\sup\{|G_0|\mid G_0\subset G\) is half-factorial\(\}\in\mathbb{N}\cup\{\infty\}\). In particular, upper and lower bounds for \(\mu(G)\) are computed, and half-factorial subsets of finite cyclic groups are characterized in terms of splittable sets.