On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid

This work introduces the first in-depth study of h-free and h-full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let m be an element of an abelian monoid, with ω(m) denoting the number of distinct prime elements generating m. We study the moments of ω(m) over subsets of h-free and h-full elements, establishing the normal order of ω(m) within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.