Suppose is a partial ordering on a nonempty set . Then the elements are said to be comparable provided or .

Because two elements in a partially ordered set need not be comparable, it is possible for a partially ordered set to have more than one maximal element. For example, suppose we have a nonempty partially ordered set in which every element is incomparable to every other element, i.e., is totally unordered. It follows that every element of is maximal.