Total Order - Strict Total Order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict total order, which can equivalently be defined in two ways:

• a < b if and only if a ≤ b and a ≠ b
• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.
• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.
• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b
• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.

We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.