# Definition:Ordered Structure

## Definition

An **ordered structure** $\struct {S, \circ, \preceq}$ is an algebraic system such that:

- $(1): \quad \struct {S, \circ}$ is an algebraic structure
- $(2): \quad \struct {S, \preceq}$ 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 $\struct {S, \circ, \preceq}$ such that $\struct {S, \circ}$ 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 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 $\struct {G, \circ, \preceq}$ such that $\struct {G, \circ}$ is a group.

### Ordered Subgroup

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

### Join Semilattice

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

- $a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.

Then the ordered structure $\struct {S, \vee, \preceq}$ is called a **join semilattice**.

### Meet Semilattice

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

- $a \wedge b \in S$,

where $a \wedge b$ is the meet of $a$ and $b$.

Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a **meet semilattice**.

### Lattice

Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.

Then $\struct {S, \vee, \wedge, \preceq}$ is called a **lattice** if and only if:

- $\struct {S, \vee, \preceq}$ is a join semilattice
- $\struct {S, \wedge, \preceq}$ is a meet semilattice

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.

- Results about
**ordered structures**can be found**here**.