Partial orders: compare what can be compared

A partial order is reflexive, antisymmetric, and transitive. Antisymmetry means that if a precedes b and b precedes a, then they must be the same object. It does not mean that reversing a comparison is always false, since reflexivity requires each object to relate to itself. It also does not require every pair to be comparable. A total order adds that additional requirement, as ordinary numerical order does for real numbers.

Subset inclusion is a model example: every set contains itself, two containments in opposite directions give equality, and nested containments compose. Two distinct singleton sets are incomparable because neither contains the other. Divisibility on positive integers is another partial order, but restricting to positive integers matters: on all nonzero integers, both 2 divides −2 and −2 divides 2 despite unequal elements. A domain choice can determine whether an axiom holds.

For a finite partial order, b covers a when a is strictly below b and no element lies strictly between them. A Hasse diagram draws just these cover edges, with larger elements higher on the page. It omits self-loops and comparisons already forced by an upward path. You recover the full order by allowing a zero-length path for equality and an upward path for every strict comparison. Edges are not arbitrary visual connections.

Consider subsets of a two-element set ordered by inclusion. The empty set is at the bottom, the two singletons are in the middle, and the whole set is at the top. There is no middle edge between the singletons because they are incomparable. There is also no direct bottom-to-top edge because a singleton lies in between. The missing direct edge does not mean the bottom and top are incomparable; the upward paths encode their comparison.

An element is maximal if no distinct element lies above it. It is greatest if every element lies below it. A greatest element is maximal and is unique by antisymmetry, but a maximal element need not dominate incomparable neighbors. Minimal and least reverse the directions. Among the subsets of a two-element set after deleting the whole set, both singletons are maximal, while neither is greatest because the other is not below it.

Every nonempty finite partially ordered set has a maximal element: keep moving strictly upward whenever possible, and finiteness prevents an endless chain of distinct elements. The argument does not prove there is only one. Nor does it apply unchanged to infinite orders: the natural numbers have no maximal element. When interpreting prerequisites or resource dependencies, a drawing's height is only a layout choice; the relation and its paths establish the mathematical comparisons.