Pseudocomplemented Lattice is Bounded
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Theorem
Let $\struct {L, \wedge, \vee, \preceq}$ be a pseudocomplemented lattice.
Then $\struct {L, \wedge, \vee, \preceq}$ is a bounded lattice.
Proof
By the definition of pseudocomplemented lattice, $L$ has a smallest element $\bot$.
Let $x \in L$.
Then:
- $x \wedge \bot = \bot$
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By the definition of pseudocomplemented lattice, there is a greatest element $\bot^*$ such that:
- $\bot \wedge \bot^* = \bot$
But then by the definition of greatest element:
- $\forall x \in L: x \preceq \bot^*$
Hnce.o $\bot^*$ is the greatest element of $L$.
Since $L$ has a smallest element and a greatest element, it is a bounded lattice.
$\blacksquare$