structural:summarization, quality:correction
Strict and non-strict total orders

A strict total order on a set X {\displaystyle X} is a strict partial order on X {\displaystyle X} in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X}, which satisfies the following for all a, b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X}:

Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry.

For delimitation purposes, a total order as defined above is sometimes called non-strict order.For each (non-strict) total order ≤ {\displaystyle \leq } there is an associated relation < {\displaystyle <}, called the strict total order associated with ≤ {\displaystyle \leq } that can be defined in two equivalent ways:

Conversely, the reflexive closure of a strict total order < {\displaystyle <} is a (non-strict) total order.