Pseudocomplemented Lattice is Bounded

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.

$\blacksquare$