Definition:Totally Ordered Set

Definition
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.

Also defined as
Different authors refer to partially ordered sets 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.

Also known as
A totally ordered set is also called a simply ordered set or linearly ordered set.

Some sources use the term chain, but this word is generally restricted to mean specifically a totally ordered subset of a given (partially) ordered set.

Also see

 * Definition:Poset
 * Definition:Well-Ordered Set


 * Definition:Chain (Set Theory)

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.