Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
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Theorem
Let $L$ be a distributive lattice.
Full notational specification needed
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Let $F$ be a filter in $L$.
Let $M$ be an ideal in $L$ which is disjoint from $F$ such that no ideal in $L$ larger than $M$ is disjoint from $F$.
Then $M$ is a prime ideal.
Proof
Suppose, for the sake of contradiction, that $M$ is not a prime ideal.
Then by Prime Ideal in Lattice, there are elements $a$ and $b$ of $L$ such that
- $a \wedge b \in M$
- $a \notin M$
- $b \notin M$
Lemma
There do not exist $m$ and $n$ in $M$ such that $m \vee a \in F$ and $n \vee b \in F$.
Proof
Suppose such exist.
Since:
- $m \vee \left({n \vee b}\right) \ge n \vee b$
- $n \vee b \in F$
- $F$ is a filter
it follows that:
- $m \vee \left({n \vee b}\right) \in F$
Applying associativity yields:
- $\left({m \vee n}\right) \vee b \in F$
By the same argument:
- $\left({m \vee n}\right) \vee a \in F$
By the definition of a filter:
- $\left({\left({m \vee n}\right) \vee b}\right) \wedge \left({\left({n \vee m}\right) \vee a}\right) \in F$
Distributing:
- $\left({m \vee n}\right) \wedge \left({b \vee a}\right) \in F$
But by assumption:
- $b \vee a \in M$
and by the definition of an ideal:
- $m \vee n \in M$
so again by the definition of an ideal:
- $\left({m \vee n}\right) \wedge \left({b \vee a}\right) \in M$
contradicting the supposition that $M$ is disjoint from $F$.
$\Box$
Without loss of generality, we can thus suppose that:
- $m \vee a \notin F$ for all $m \in M$
Let $N = \left\{ {x \in L: x \le m \vee a \text{ for some } m \in M}\right\}$.
Lemma
$N$ is an ideal in $L$.
Proof
Let:
- $x \in N$
- $y \in L$
- $y \le x$
Then by the definition of $N$ there exists an $m \in M$ such that:
- $x \le m \vee a$
Since $y \le x$, it follows that:
- $y \le m \vee a$
so $y \in N$.
Let:
- $x \in N$
- $y \in N$
Then there exist $m_x$ and $m_y$ in $M$ such that:
- $x \le m_x \vee a$
- $y \le m_y \vee a$
Then:
- $x \vee y \le \left({m_x \vee a}\right) \vee \left({m_y \vee a}\right) = \left({m_x \vee m_y}\right) \vee a$
But $m_x \vee m_y \in M$, so:
- $x \vee y \in N$
$\Box$
Lemma
$M \subsetneq N$
Proof
Let $m \in M$.
Then:
- $m \le \left({m \vee a}\right)$
so $m \in N$.
Thus $M \subseteq N$.
We have:
- $a \le \left({m \vee a}\right)$
so:
- $a \in N$
but:
- $a \notin M$
Thus $M \subsetneq N$.
$\Box$
Lemma
$N \cap F = \varnothing$
Proof
Suppose $x \in N \cap F$.
Then:
- $x \in N$
so for some $m \in M$:
- $x \le m \vee a$
Furthermore, $x \in F$.
So by the definition of a filter:
- $m \vee a \in F$
But this contradicts our assumption that $m \vee a \notin F$ for all $m \in M$.
$\Box$
By assuming that $M$ is not a prime ideal, we have constructed an ideal $N$ properly containing $M$ that is disjoint from $F$, contradicting the maximality of $M$.
Thus $M$ is a prime ideal.
$\blacksquare$