Definition:Total Ordering

Let $$\left({S; \preceq}\right)$$ be a poset.

Then the ordering $$\preceq$$ is a total ordering on $$S$$ iff $$\left({S; \preceq}\right)$$ has no non-comparable pairs:


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

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

If this is the case, then $$\left({S; \preceq}\right)$$ is referred to as a totally ordered set or toset.

Some sources call this a linear ordering, or a simple ordering.

Weak vs. Strict Orderings
Compare strict total ordering.

If it is necessary to emphasise that a total ordering $$\preceq$$ is not strict, then the term weak total ordering may be used.