Editing Preorder (section)

==Relationship to strict partial orders==

{{anchor|Strict preorder}}
If reflexivity is replaced with [[Irreflexive relation|irreflexivity]] (while keeping transitivity) then we get the definition of a [[strict partial order]] on <math>P</math>. For this reason, the term '''{{em|strict preorder}}''' is sometimes used for a strict partial order. That is, this is a binary relation <math>\,<\,</math> on <math>P</math> that satisfies:
<ol>
<li>[[Irreflexive relation|Irreflexivity]] or anti-reflexivity: {{em|not}} <math>a < a</math> for all <math>a \in P;</math> that is, <math>\,a < a</math> is {{em|false}} for all <math>a \in P,</math> and</li>
<li>[[Transitive relation|Transitivity]]: if <math>a < b \text{ and } b < c \text{ then } a < c</math> for all <math>a, b, c \in P.</math></li>
</ol>

===Strict partial order induced by a preorder===

Any preorder <math>\,\lesssim\,</math> gives rise to a strict partial order defined by <math>a < b</math> if and only if <math>a \lesssim b</math> and not <math>b \lesssim a</math>.
Using the equivalence relation <math>\,\sim\,</math> introduced above, <math>a < b</math> if and only if <math>a \lesssim b \text{ and not } a \sim b;</math> 
and so the following holds
<math display=block>a \lesssim b \quad \text{ if and only if } \quad a < b \; \text{ or } \; a \sim b.</math>
The relation <math>\,<\,</math> is a [[strict partial order]] and {{em|every}} strict partial order can be constructed this way. 
{{em|If}} the preorder <math>\,\lesssim\,</math> is [[Antisymmetric relation|antisymmetric]] (and thus a partial order) then the equivalence <math>\,\sim\,</math> is equality (that is, <math>a \sim b</math> if and only if <math>a = b</math>) and so in this case, the definition of <math>\,<\,</math> can be restated as: 
<math display=block>a < b \quad \text{ if and only if } \quad a \lesssim b \; \text{ and } \; a \neq b \quad\quad (\text{assuming } \lesssim \text{ is antisymmetric}).</math>
But importantly, this new condition is {{em|not}} used as (nor is it equivalent to) the general definition of the relation <math>\,<\,</math> (that is, <math>\,<\,</math> is {{em|not}} defined as: <math>a < b</math> if and only if <math>a \lesssim b \text{ and } a \neq b</math>) because if the preorder <math>\,\lesssim\,</math> is not antisymmetric then the resulting relation <math>\,<\,</math> would not be transitive (consider how equivalent non-equal elements relate). 
This is the reason for using the symbol "<math>\lesssim</math>" instead of the "less than or equal to" symbol "<math>\leq</math>", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that <math>a \leq b</math> implies <math>a < b \text{ or } a = b.</math> 

===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>