The relation

divides

is a partial order: it is reflexive, antisymmetric, and transitive. These properties are expressed in the directed graphs generated here. Reflexivity: there is a path of (zero) arrows from each vertex to itself. Antisymmetry: if there is a path of arrows from

to

, there cannot be a path in the reverse direction. Transitivity: a path of arrows from

to

can be appended to a path of arrows from

to

to create a path from

to

.