# Total Strict Order Relation

A Total Strict Order Relation is a Binary Relation that is a Transitive, a Antisymmetric and a Semiconnex Relation.

## References

### 2020a

• (Wikipedia, 2020a) ⇒ https://en.wikipedia.org/wiki/Total_order Retrieved:2020-2-15.
• In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some set $X$ , which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a chain,a totally ordered set,a simply ordered set,or a linearly ordered set. Formally, a binary relation $\leq$ is a total order on a set $X$ if the following statements hold for all $a, b$ and $c$ in $X$ :
• Antisymmetry: If $a \leq b$ and $b \leq a$ then $a = b$ ;
• Transitivity: If $a \leq b$ and $b \leq c$ then $a \leq c$ ;
• Connexity: $a \leq b$ or $b \leq a$ .
(...)

### 2020b

• (Wikipedia, 2020b) ⇒ https://en.wikipedia.org/wiki/Total_order#Strict_total_order Retrieved:2020-2-15.
• For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) transitive semiconnex relation <, called a strict total order or strict semiconnex order,which can be defined in two equivalent ways:
• a < b if ab and ab
• a < b if not ba (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 be defined in two equivalent ways:
• ab if a < b or a = b
• ab 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.