202309121209

Tags : Logic

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Filters in Heyting Algebras

A Filter in a Heyting Algebra $\mathcal{H}=⟨H,\bigsqcup ,\sqcap ,\Rightarrow ,1,0⟩$ is a non empty subset $F$ of $H$ such that

• $a,b\in F$ implies $a\sqcap b\in F$
• $a\in F$ and $a\le b$ implies $b\in F$

$F$ is Proper iff $F⊊H$.

$F$ is called a prime filter if $a\bigsqcup b\in F$ implies $a\in F$ or $b\in F$

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References

• Filters
• Extension of Proper Filter to Prime Filter in Heyting Algebras