# Definition:Bounded Lattice/Definition 2

## Definition

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.

Thus $\left({S, \vee, \wedge}\right)$ is a bounded lattice iff the following axioms are satisfied:

 $(L0)$ $:$ Closure $\displaystyle \forall a, b:$ $\displaystyle a \vee b \in S$ $\displaystyle a \wedge b \in S$ $(L1)$ $:$ Commutativity $\displaystyle \forall a, b:$ $\displaystyle a \vee b = b \vee a$ $\displaystyle a \wedge b = b \wedge a$ $(L2)$ $:$ Associativity $\displaystyle \forall a, b, c:$ $\displaystyle a \vee \left({b \vee c}\right) = \left({a \vee b}\right) \vee c$ $\displaystyle a \wedge \left({b \wedge c}\right) = \left({a \wedge b}\right) \wedge c$ $(L3)$ $:$ Idempotence $\displaystyle \forall a:$ $\displaystyle a \vee a = a$ $\displaystyle a \wedge a = a$ $(L4)$ $:$ Absorption $\displaystyle \forall a,b:$ $\displaystyle a \vee \left({a \wedge b}\right) = a$ $\displaystyle a \wedge \left({a \vee b}\right) = a$ $(L5)$ $:$ Identity elements $\displaystyle \exists \top, \bot: \forall a:$ $\displaystyle a \vee \bot = a = \bot \vee a$ $\displaystyle a \wedge \top = a = \top \vee a$