Editing Total order (section)

==Strict and non-strict total orders==

For delimitation purposes, a total order as defined [[#Top|above]] is sometimes called ''non-strict'' order.
For each (non-strict) total order <math>\leq</math> there is an associated relation <math><</math>, called the ''strict total order'' associated with <math>\leq</math> that can be defined in two equivalent ways:
* <math>a < b</math> if <math>a \leq b</math> and <math>a \neq b</math> ([[reflexive reduction]]).
* <math>a < b</math> if not <math>b \leq a</math> (i.e., <math><</math> is the [[Binary relation#Complement|complement]] of the [[converse relation|converse]] of <math>\leq</math>).

Conversely, the [[reflexive closure]] of a strict total order <math><</math> is a (non-strict) total order.

Thus, a '''{{em|strict total order}}''' on a set <math>X</math> is a [[strict partial order]] on <math>X</math> in which any two distinct elements are comparable. That is, a strict total order is a [[binary relation]] <math><</math> on some [[Set (mathematics)|set]] <math>X</math>, which satisfies the following for all <math>a, b</math> and <math>c</math> in <math>X</math>:
# Not <math>a < a</math> ([[Irreflexive relation|irreflexive]]).
# If <math>a < b</math> then not <math> b < a </math> ([[asymmetric relation|asymmetric]]).
# If <math>a < b</math> and <math>b < c</math> then <math>a < c</math> ([[Transitive relation|transitive]]).
# If <math>a \neq b</math>, then <math>a < b</math> or <math>b < a</math> ([[Connected relation|connected]]).

Asymmetry follows from transitivity and irreflexivity;<ref>Let <math>a < b</math>, assume for contradiction that also <math> b < a </math>. Then <math>a < a</math> by transitivity, which contradicts irreflexivity.</ref> moreover, irreflexivity follows from asymmetry.<ref>If <math>a < a</math>, the not <math>a < a</math> by asymmetry.</ref>