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.