divisorie

Divisorie is a term used in mathematical discussions to denote a structured representation of the divisibility relations among the positive divisors of an integer, or more generally a framework for analyzing divisibility within a partially ordered set. The concept is closely related to the study of divisor lattices and poset representations.

Formally, for a positive integer n, the divisorie of n is the directed acyclic graph whose nodes are the divisors of n, with a directed edge d1 -> d2 whenever d1 divides d2 and there is no divisor e with d1 < e < d2 among the divisors of n. Equivalently, it can be viewed as the Hasse diagram of the divisor lattice D(n) ordered by divisibility. The structure highlights how divisors build up from 1 to n through multiplicative steps and clarifies the relationships among factors.

Example: for n = 12, the divisors are 1, 2, 3, 4, 6, 12. The edges (1->2), (1->3), (2->4), (2->6), (3->6), (4->12), (6->12) form the divisorie. The diagram emphasizes prime-power decompositions and the ways factors combine to form n.

Variants of the divisorie include weighted versions that assign values to edges reflecting the ratio of consecutive divisors, or layered representations that group divisors by their prime-power structure. In abstract settings, divisorie concepts can be extended to the divisibility relations in sets of integers, gcd-lcm lattices, or polynomial rings.

See also: divisor function, divisor lattice, Hasse diagram, poset, lattice theory.