Complete Lattice has Both Greatest Element and Smallest Element
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Theorem
Let $\struct{S, \vee, \wedge, \preceq}$ be a complete lattice.
Then:
- $(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:
- $\quad \bot := \sup \O$
- $(\text{2}) \quad \struct{S, \preceq}$ has a greatest element, namely:
- $\quad \top := \inf \O$
Proof
From Complete Lattice is Bounded:
- $\struct{S, \vee, \wedge, \preceq}$ is a bounded lattice
From Bounded Lattice has Both Greatest Element and Smallest Element:
- $(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:
- $\quad \bot := \sup \O$
- $(\text{2}) \quad \struct{S, \preceq}$ has a greatest element, namely:
- $\quad \top := \inf \O$
$\blacksquare$