Chapter 51: Set-Theoretic Operations

Having defined membership as divisibility , we now earn the standard set-theoretic operations — union, intersection, and set difference. The key insight: union is least common multiple, intersection is greatest common divisor, and difference is exponent-wise subtraction in prime factorizations. These operations endow τ-Idx with the structure of a distributive lattice, with 1 as the bottom element and no finite top. Every set-theoretic computation reduces to an arithmetic operation on prime factorizations.