Definition:Totally Ordered Set

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

Then $$\left({S; \preceq}\right)$$ is a totally ordered set if $$\preceq$$ is a total ordering.

Different authors refer to partially and totally ordered sets in different ways. Some use "ordering" to mean a "partial ordering", while others use "ordering" to mean a "total ordering".

As totally ordered sets are still posets, all results applying to posets also apply to (totally) ordered sets. So it is usual to use the term "poset" to mean a set which may be partially or totally ordered, and "partially ordered set" when we want to make it clear that the set under discussion is definitely not totally ordered.