Definition:Bounded Lattice/Definition 2
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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 | \(\displaystyle \exists \top, \bot: \forall a:\) | \(\displaystyle a \vee \bot = a = \bot \vee a \) | \(\displaystyle a \wedge \top = a = \top \vee a \) |
