Factoriality inside Boolean lattices

Given a join semilattice $S$ with a minimum $\hat{0}$, the quarks (also called atoms in order theory) are the elements that cover $\hat{0}$, and for each $x \in S \setminus \{\hat{0}\}$ a factorization (into quarks) of $x$ is a minimal set of quarks whose join is $x$. If every element $x \in S \setminus \{\hat{0}\}$ has a factorization, then $S$ is called factorizable. If for each $x \in S \setminus \{\hat{0}\}$, any two factorizations of $x$ have equal (resp., distinct) size, then we say that $S$ is half-factorial (resp., length-factorial). Let $B_\mathbb{N}$ be the Boolean lattice consisting of all finite subsets of $\mathbb{N}$ under intersections and unions. Here we study factorizations into quarks in join subsemilattices of $B_\mathbb{N}$, focused on the notions of half-factoriality and length-factoriality. We also consider the unique factorization property, which is the most special and relevant type of half-factoriality, and the elasticity, which is an arithmetic statistic that measures the deviation from half-factoriality.