Definition:Strict Total Ordering

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:


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

That is, iff $$\prec$$ is connected.

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:


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


 * $$a \prec b \iff \neg b \preceq a$$.

This is proved in Complement of Strict Total Ordering.