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.

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