# Definition:Bounded Lattice/Definition 2

## 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 bounded lattice axioms are satisfied:

 $(\text L 0)$ $:$ Closure $\ds \forall a, b \in S:$ $\ds a \vee b \in S$ $\ds a \wedge b \in S$ $(\text L 1)$ $:$ Commutativity $\ds \forall a, b \in S:$ $\ds a \vee b = b \vee a$ $\ds a \wedge b = b \wedge a$ $(\text L 2)$ $:$ Associativity $\ds \forall a, b, c \in S:$ $\ds a \vee \paren {b \vee c} = \paren {a \vee b} \vee c$ $\ds a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c$ $(\text L 3)$ $:$ Idempotence $\ds \forall a \in S:$ $\ds a \vee a = a$ $\ds a \wedge a = a$ $(\text L 4)$ $:$ Absorption $\ds \forall a, b \in S:$ $\ds a \vee \paren {a \wedge b} = a$ $\ds a \wedge \paren {a \vee b} = a$ $(\text L 5)$ $:$ Identity elements $\ds \exists \top, \bot \in S: \forall a \in S:$ $\ds a \vee \bot = a = \bot \vee a$ $\ds a \wedge \top = a = \top \wedge a$

## Also see

• Results about bounded lattices can be found here.