Definition:Strict Total Ordering

Definition
Let $\struct {S, \prec}$ be a relational structure.

Let $\prec$ be a strict ordering.

Then $\prec$ is a strict total ordering on $S$ $\struct {S, \prec}$ 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
Other terms in use are simple order and order relation.

Some sources, for example and, call this a linear order.

As this term is also used by other sources to mean total ordering, it is preferred that on  the terms partial, total and well, and weak and strict, are the only terms to be used to distinguish between different types of ordering.

Also see

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