Definition:Bounded Lattice/Definition 2

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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.