On the distribution of the number of distinct generators of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -free and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -f

This work introduces the first in-depth study of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -free and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔪</mml:mi> </mml:math> be an element of an abelian monoid, with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>𝔪</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denoting the number of distinct prime elements generating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔪</mml:mi> </mml:math> . We study the moments of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>𝔪</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> over subsets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -free and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -full elements, establishing the normal order of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>𝔪</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.