# Definition:Lattice (Order Theory)

*This page is about Lattice in the context of Order Theory. For other uses, see Lattice.*

## Definition

### Definition 1

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

Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima.

Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively.

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

### Definition 2

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.

### Definition 3

Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$.

Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$:

- $a \vee \paren {a \wedge b} = a$
- $a \wedge \paren {a \vee b} = a$

Let $\preceq$ be the ordering on $S$ defined by:

- $\forall a, b \in S: a \preceq b$ if and only if $a \vee b = b$

as on Semilattice Induces Ordering.

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

## Also defined as

Some sources refer to a **bounded lattice** as a **lattice**.

This comes down to insisting that $\vee$ and $\wedge$ admit identity elements.

## Also denoted as

In the particular context of order theory, it is common to omit $\vee$ and $\wedge$ in the notation for a **lattice**.

That is, one then writes $\struct {S, \preceq}$ in place of $\struct {S, \vee, \wedge, \preceq}$.

## Also known as

Some sources, in order to reduce confusion between a **lattice** and a **point lattice**, use the term **order lattice** for the former.

## Also see

- Definition:Bounded Lattice
- Definition:Join Semilattice
- Definition:Meet Semilattice
- Definition:Semilattice
- Definition:Complete Lattice
- Definition:Lattice (Ordered Set): a specific instantiation of a
**lattice**in the context of an ordered set

- Results about
**lattices**in the context of**order theory**can be found**here**.