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.