The reachability condition on a directed graph is reflexive and transitive so it produces a preorder. Symmetric preorders are equivalence relations and antisymmetric preorders are partial orders. Equivalence relations are partially ordered in terms of fineness and coarseness.
Parthood relations form a partial order. In particular, parthood relations are often hierarchical in nature so they can be modelled by undirected trees or polytrees.
Many to one relations:
Many to one relations induce an equivalence relation whose equivalence classes are the many part of the relation. Many to one relations are strong simplification rules for equivalence relations.