Editing Directed set (section)

===Maximal and greatest elements===

An element <math>m</math> of a preordered set <math>(I, \leq)</math> is a ''[[Maximal and minimal elements|maximal element]]'' if for every <math>j \in I,</math> <math>m \leq j</math> implies <math>j \leq m.</math>{{efn|This implies <math>j = m</math> if <math>(I, \leq)</math> is a [[partially ordered set]].}}
It is a ''[[Greatest element and least element|greatest element]]'' if for every <math>j \in I,</math> <math>j \leq m.</math>

Any preordered set with a greatest element is a directed set with the same preorder. 
For instance, in a [[poset]] <math>P,</math> every [[Upper set#Upper closure and lower closure|lower closure]] of an element; that is, every subset of the form <math>\{a \in P : a \leq x\}</math> where <math>x</math> is a fixed element from <math>P,</math> is directed.

Every maximal element of a directed preordered set is a greatest element.  Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal and of greatest elements.