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.

Some sources, for example W.E. Deskins: Abstract Algebra (1964), 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.

Sources

• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$: Definition $1.7 \ \text {(a)}$
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets