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.
Read more about this topic: Total Order
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