Editing Preorder (section)

===Preorders induced by a strict partial order===

Using the construction above, multiple non-strict preorders can produce the same strict preorder <math>\,<,\,</math> so without more information about how <math>\,<\,</math> was constructed (such knowledge of the equivalence relation <math>\,\sim\,</math> for instance), it might not be possible to reconstruct the original non-strict preorder from <math>\,<.\,</math> Possible (non-strict) preorders that induce the given strict preorder <math>\,<\,</math> include the following:
* Define <math>a \leq b</math> as <math>a < b \text{ or } a = b</math> (that is, take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<math><</math>" through reflexive closure; in this case the equivalence is equality <math>\,=,</math> so the symbols <math>\,\lesssim\,</math> and <math>\,\sim\,</math> are not needed. 
* Define <math>a \lesssim b</math> as "<math>\text{ not } b < a</math>" (that is, take the inverse complement of the relation), which corresponds to defining <math>a \sim b</math> as "neither <math>a < b \text{ nor } b < a</math>"; these relations <math>\,\lesssim\,</math> and <math>\,\sim\,</math> are in general not transitive; however, if they are then <math>\,\sim\,</math> is an equivalence; in that case "<math><</math>" is a [[strict weak order]]. The resulting preorder is [[Connected relation|connected]] (formerly called total); that is, a [[total preorder]].

If <math>a \leq b</math> then <math>a \lesssim b.</math> 
The converse holds (that is, <math>\,\lesssim\;\; = \;\;\leq\,</math>) if and only if whenever <math>a \neq b</math> then <math>a < b</math> or <math>b < a.</math>