Definition:Strict Total Ordering

Definition
Let $\left({S, \prec}\right)$ be a relational structure.

Let $\prec$ be a strict ordering.

Then $\prec$ is a strict total ordering on $S$ $\left({S, \prec}\right)$ has no non-comparable pairs:


 * $\forall x, y \in S: x \ne y \implies x \prec y \lor y \prec x$

That is, $\prec$ is connected.

Also known as
Some sources, for example, call this a linear order.

As this term is also used by other sources to mean total ordering, care is advised to make sure you know exactly what is being referred to.

Other terms in use are simple order and order relation

Also see

 * Definition:Weak Total Ordering
 * Complement of Strict Total Ordering