# Definition:Bounded Lattice

## Definition

### Definition 1

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

Let $S$ admit all finite suprema and finite infima.

Let $\vee$ and $\wedge$ be the join and meet operations on $S$, respectively.

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

### Definition 2

Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.

Let $\vee$ and $\wedge$ have identity elements $\bot$ and $\top$ respectively.

Then $\struct {S, \vee, \wedge, \preceq}$ is a **bounded lattice**.

### Definition 3

Let $\struct {S, \wedge, \vee, \preceq}$ be a lattice.

Let $S$ be bounded in $\struct {S, \preceq}$.

Then $\struct {S, \wedge, \vee, \preceq}$ is a **bounded lattice**.

## Notation

The **greatest element** and **smallest element** of a **bounded lattice** are denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\top$ and $\bot$ respectively.

Some sources use $1$ for the **greatest element** and $0$ for the **smallest element**.

## Also known as

Some authors insist that a lattice have identity elements, and so refer to a bounded lattice simply as a **lattice**.

## Also see

- Results about
**bounded lattices**can be found**here**.