Editing Preorder (section)

==Related definitions==

If a preorder is also [[Antisymmetric relation|antisymmetric]], that is, <math>a \lesssim b</math> and <math>b \lesssim a</math> implies <math>a = b,</math> then it is a [[Partially ordered set|partial order]].

On the other hand, if it is [[Symmetric relation|symmetric]], that is, if <math>a \lesssim b</math> implies <math>b \lesssim a,</math> then it is an [[equivalence relation]].

A preorder is [[Total preorder|total]] if <math>a \lesssim b</math> or <math>b \lesssim a</math> for all <math>a, b \in P.</math>

A [[preordered class]] is a [[Class (mathematics)|class]] equipped with a preorder. Every set is a class and so every preordered set is a preordered class.