# Definition:Bounded Lattice

## Definition

### Definition 1

Let $\left({S, \preceq}\right)$ 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 $\left({S, \vee, \wedge, \preceq}\right)$ is a bounded lattice.

### Definition 2

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

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

Then $\left({S, \vee, \wedge, \preceq}\right)$ is a bounded lattice.

### Definition 3

Let $\left({S, \wedge, \vee, \preceq}\right)$ be a lattice.

Let $S$ be bounded in $\left({S,\preceq}\right)$.

Then $\left({S, \wedge, \vee, \preceq}\right)$ is a bounded lattice.

## Equivalence of Definitions

These definitions are shown to be equivalent in Equivalence of Definitions of Bounded Lattice.

## 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.