Definition:Totally Ordered Structure

Definition
Let $\left({S, \circ, \preceq}\right)$ be an ordered structure.

That is:
 * $(1): \quad \left({S, \circ}\right)$ is an algebraic structure
 * $(2): \quad \left({S, \preceq}\right)$ is an ordered set
 * $(3): \quad \preceq$ is compatible with $\circ$.

When the ordering $\preceq$ is a total ordering, the structure $\left({S, \circ, \preceq}\right)$ is then a totally ordered structure.