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.