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.
Totally Ordered Semigroup
A totally ordered semigroup is a totally ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.
Totally Ordered Commutative Semigroup
A totally ordered commutative semigroup is a totally ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative semigroup.
Totally Ordered Group
A totally ordered group is a totally ordered structure $\left({G, \circ, \preceq}\right)$ such that $\left({G, \circ}\right)$ is a group.
Totally Ordered Ring
Let $\struct {R, +, \circ, \preceq}$ be an ordered ring.
Let the ordering $\preceq$ be a total ordering.
Then $\struct {R, +, \circ, \preceq}$ is a totally ordered ring.
Totally Ordered Field
Let $\struct {F, +, \circ, \preceq}$ be an ordered ring.
Let $\struct {F, +, \circ}$ be a field.
Let the ordering $\preceq$ be a total ordering.
Then $\struct {F, +, \circ, \preceq}$ is a totally ordered field.