or partial ordering, n. a relation that is reflexive, antisymmetric and transitive, but not necessarily connected; this imposes a lattice on a set. A partial ordering generates chains of elements, but unless it is a total ordering there will be pairs of elements in the set between which the relation does not hold in either order; for example, in the tree diagram below, where x ≤ y if and only if there is a path from the origin to y via x, clearly it is neither the case that A ≤ B, nor B ≤ A. In particular, although each finite chain will have both a maximal and a minimal element, these may not be unique. For example, the subsets of the integers have a partial ordering under set inclusion; in this case there is a unique minimal element, the empty set, that is contained in all members of the domain, and a unique maximal element, the set of integers itself, that contains all elements of the domain, but not all pairs of elements are related, as neither of {1, 2, 3} and {2, 3, 4} contains the other. If the domain is restricted to non-empty proper subsets, and the ordering is strict inclusion, there are infinitely many maximal and minimal elements. See also poset, ordering, Zorn's lemma.