Definition:Totally Ordered Set

Definition
Let $\left({S, \preceq}\right)$ be a relational structure.

Then $\left({S, \preceq}\right)$ is a totally ordered set $\preceq$ 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.

It is also known as a toset. This term may be encountered on.

Some sources refer to a totally ordered set as an ordered set, using the term partially ordered set for what goes as an ordered set on.

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

Also see

 * Definition:Partially Ordered Set
 * Definition:Well-Ordered Set


 * Definition:Chain (Set Theory)

As totally ordered sets are still ordered sets, all results applying to ordered sets also apply to totally ordered sets.

So it is usual to use the term ordered set 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.