Editing Weak ordering (section)

===Total preorders<!--This section is linked from [[Preference (economics)]]-->===

Strict weak orders are very closely related to '''total preorders''' or '''(non-strict) weak orders''', and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a [[preorder]] in which any two elements are comparable.<ref>Such a relation is also called [[Connected relation|strongly connected]].</ref> A total preorder <math>\,\lesssim\,</math> satisfies the following properties:

* {{em|Transitivity}}: For all <math>x, y, \text{ and } z,</math> if <math>x \lesssim y \text{ and } y \lesssim z</math> then <math>x \lesssim z.</math>
* {{em|Strong connectedness}}: For all <math>x \text{ and } y,</math> <math>x \lesssim y \text{ or } y \lesssim x.</math>
** Which implies {{em|reflexivity}}: for all <math>x,</math> <math>x \lesssim x.</math>

A [[total order]] is a total preorder which is antisymmetric, in other words, which is also a [[Partially ordered set|partial order]]. Total preorders are sometimes also called '''preference relations'''.

The [[Complement (set theory)|complement]] of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements.  Thus we take the [[Converse relation|converse]] of the complement: for a strict weak ordering <math>\,<,</math> define a total preorder <math>\,\lesssim\,</math> by setting <math>x \lesssim y</math> whenever it is not the case that <math>y < x.</math> In the other direction, to define a strict weak ordering < from a total preorder <math>\,\lesssim,</math> set <math>x < y</math> whenever it is not the case that <math>y \lesssim x.</math><ref>{{citation|title=Multicriteria Optimization|first=Matthias|last=Ehrgott|publisher=Springer|year=2005|isbn=9783540276593|url=https://books.google.com/books?id=AwRjo6iP4_UC&pg=PA10|at=Proposition 1.9, p.&nbsp;10}}.</ref>

In any preorder there is a [[Preorder#Constructions|corresponding equivalence relation]] where two elements <math>x</math> and <math>y</math> are defined as '''equivalent''' if <math>x \lesssim y \text{ and } y \lesssim x.</math> In the case of a {{em|total}} preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering.