Definition:Bounded Lattice

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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.