Editing Preorder (section)

===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>