Definition:Totally Ordered Structure

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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.