Definition:Totally Ordered Set

Definition
Let $\struct {S, \preceq}$ be a relational structure.

Then $\struct {S, \preceq}$ is a totally ordered set $\preceq$ is a total ordering.

Totally Ordered Class
The concept carries naturally over into class theory:

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.

The term permutation is an older term for totally ordered set, but has since been changed to mean the bijection that can be applied on such a totally ordered set in order to redefine its ordering.

Also see

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


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


 * Definition:Chain (Set Theory)