Definition:Totally Ordered Set

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

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

Different authors refer to partially and totally ordered sets in different ways. Some use "ordering" to mean an ordering which may or may not be partial, while others use "ordering" exclusively to mean a total ordering.

As tosets are still posets, all results applying to posets also apply to tosets.

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.