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 $

\((L0)\)	$:$	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	\(\displaystyle \exists \top, \bot: \forall a:\)	\(\displaystyle a \vee \bot = a = \bot \vee a \)	\(\displaystyle a \wedge \top = a = \top \vee a \)

Definition:Bounded Lattice/Definition 2

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