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:

$(1): \quad \struct {S, \vee, \preceq}$ is a join semilattice

and:

$(2): \quad \struct {S, \wedge, \preceq}$ 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.