Characterization of Locale/Statement 3 Implies Statement 4
Jump to navigation
Jump to search
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Theorem
Let $L = \struct{S, \preceq}$ be an complete lattice satisfying the infinite join distributive law.
Then:
- $L$ is a Heyting algebra
Proof
Let $a, b \in S$.
Let $a \to b = \sup \set{c \in S : a \wedge c \preceq b}$
We have:
\(\ds a \wedge \paren{a \to b}\) | \(=\) | \(\ds a \wedge \sup \set{c : a \wedge c \preceq b}\) | Definition of $a \to b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sup \set{a \wedge c : a \wedge c \preceq b}\) | Infinite join distributive law | |||||||||||
\(\ds \) | \(\preceq\) | \(\ds b\) | Definition of Supremum of Set |
Hence:
- $a \to b$ is the greatest element $c$ such that:
- $c \wedge a \preceq b$
It follows that $a \to b$ is a relative psuedocomplement by definition.
Since $a, b$ were arbitrary, then:
- $\forall a, b \in S : \exists a \to b \in S : a \to b$ is the greatest element $c \in L$ such that $a \wedge c \preceq b$
Hence $L$ is a Heyting algebra by definition.
$\blacksquare$