# Definition:Bounded Lattice/Definition 2

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## Definition

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**.

Thus $\struct {S, \vee, \wedge, \preceq}$ is a **bounded lattice** if and only if the following axioms are satisfied:

\((L0)\) | $:$ | Closure | \(\ds \forall a, b \in S:\) | \(\ds a \vee b \in S \) | \(\ds a \wedge b \in S \) | |||

\((L1)\) | $:$ | Commutativity | \(\ds \forall a, b \in S:\) | \(\ds a \vee b = b \vee a \) | \(\ds a \wedge b = b \wedge a \) | |||

\((L2)\) | $:$ | Associativity | \(\ds \forall a, b, c \in S:\) | \(\ds a \vee \paren {b \vee c} = \paren {a \vee b} \vee c \) | \(\ds a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c \) | |||

\((L3)\) | $:$ | Idempotence | \(\ds \forall a \in S:\) | \(\ds a \vee a = a \) | \(\ds a \wedge a = a \) | |||

\((L4)\) | $:$ | Absorption | \(\ds \forall a, b \in S:\) | \(\ds a \vee \paren {a \wedge b} = a \) | \(\ds a \wedge \paren {a \vee b} = a \) | |||

\((L5)\) | $:$ | Identity elements | \(\ds \exists \top, \bot \in S: \forall a \in S:\) | \(\ds a \vee \bot = a = \bot \vee a \) | \(\ds a \wedge \top = a = \top \wedge a \) |