# Definition:Ordered Structure

## Contents

## Definition

An **ordered structure** $\left({S, \circ, \preceq}\right)$ is a set $S$ such that:

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

There are various breeds of ordered structure the same way that there are for algebraic structures:

### Ordered Semigroup

An **ordered semigroup** is an **ordered structure** $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.

### Ordered Commutative Semigroup

An **ordered commutative semigroup** is an ordered semigroup $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a commutative semigroup.

### Ordered Subsemigroup

An **ordered subsemigroup** $\left({T, \circ, \preceq}\right)$ of an **ordered structure** $\left({S, \circ, \preceq}\right)$ is an ordered semigroup such that the semigroup $\left({T, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.

### Ordered Monoid

An **ordered monoid** is an **ordered structure** $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a monoid.

### Ordered Group

An **ordered group** is an **ordered structure** $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a group.

### Ordered Subgroup

An **ordered subgroup** $\left({T, \circ, \preceq}\right)$ of an **ordered structure** $\left({S, \circ, \preceq}\right)$ is an ordered group such that the group $\left({T, \circ}\right)$ is a subgroup of $\left({S, \circ}\right)$.

The list goes on; we won't labour the point.

## Totally Ordered Structure

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

As above, this has its various sub-breeds.

## Also known as

In order to reduce confusion with the concept of an ordered set, an **ordered structure** is sometimes referred to as an **ordered algebraic structure**.

## Also see

- Ordered Set: this is also sometimes referred to as an
**ordered structure**, or sometimes an**order structure**, on the grounds that it is a relational structure which happens to be an ordering.

- Ordered Ring, in which the definition is subtly different.