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 $S$ iff $\left({S, \prec}\right)$ has no non-comparable pairs:


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

That is, iff $\prec$ 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.

Weak vs. Strict Orderings
An alternative way of defining a strict total ordering is as follows.

For each (weak) ordering relation $\preceq$, there is an associated strict total ordering relation $\prec$, which can be defined in either of two ways:


 * $a \prec b \iff a \preceq b \land a \ne b$;


 * $a \prec b \iff \neg \left({b \preceq a}\right)$.

This is proved in Complement of Strict Total Ordering.