# If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter

## Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.

Let $I$ be an ideal in $L$.

Let $F$ be a filter on $L$ such that

- $I \cap F = \O$

Then there exists a prime ideal $P$ in $L$:
$I \subseteq P$ and $P \cap F = \O$

## Proof

Define $X := \set {P \in \mathit{Ids}\left({L}\right): I \subseteq P \land P \cap F = \O}$

where $\mathit{Ids}\left({L}\right)$ denotes set of all ideals in $L$.

- $I \in X$

We will prove that

- $\forall Z: Z \ne \varnothing \land Z \subseteq X \land \left({\forall Y_1, Y_2 \in Z: Y_1 \subseteq Y_2 \lor Y_2 \subseteq Y_1}\right) \implies \bigcup Z \in X$

Let $Z$ such that

- $Z \ne \varnothing$ and
- $Z \subseteq X$ and
- $\forall Y_1, Y_2 \in Z: Y_1 \subseteq Y_2 \lor Y_2 \subseteq Y_1$

By definition of $X$:

By definition of $X$:

- $\forall Y \in Z: Y$ is lower set.

Then by Every Element is Lower implies Union is Lower:

- $J := \bigcup Z$ is lower set.

By definition of non-empty set:

- $\exists Y: Y \in Z$

By definition of subset:

- $Y \in X$

By definition of $X$:

- $I \subseteq Y$

By Set is Subset of Union/General Result:

- $Y \subseteq J$

By Subset Relation is Transitive:

- $I \subseteq J$

We will prove that

- $\forall A, B \in Z: \exists C \in Z: A \cup B \subseteq C$

Let $A, B \in Z$.

By assumption:

- $A \subseteq B$ or $B \subseteq A$

By Union with Superset is Superset:

- $A \cup B = B$ or $A \cup B = A$

Thus by Set is Subset of Itself:

- $\exists C \in Z: A \cup B \subseteq C$

$\Box$

By definition of $X$:

- $\forall Y \in Z: Y$ is directed.

- $J$ is directed.

By definition of ideal in ordered set:

- $J$ is an ideal in $L$.

We will prove that

- $J \cap F = \varnothing$

Let $x \in J$.

By definition of union:

- $\exists A \in Z: x \in A$

By definition of subset:

- $A \in X$

By definition of $X$:

- $A \cap F = \varnothing$

Thus by definitions of intersection and empty set:

- $x \notin F$

$\Box$

Thus by definition of $X$:

- $\bigcup Z \in X$

$\Box$

By Zorn's Lemma:

- there exists $Y \in X$: $Y$ is maximal set of $X$.

Then by definition of $X$:

- $Y \in \mathit{Ids}\left({L}\right)$ and $I \subseteq Y$ and $Y \cap F = \varnothing$

We will prove that

- $Y$ is a prime ideal.

Let $x, y \in S$ such that

- $x \wedge y \in Y$

Aiming for a contradiction suppose that

- $x \notin Y$ and $y \notin Y$

Define $P_y = \operatorname{finsups}\left({Y \cap \left\{ {y}\right\} }\right)^\preceq$

By Finite Suprema Set and Lower Closure is Smallest Ideal:

- $Y \cup \left\{ {y}\right\} \subseteq P_y$

- $Y \subseteq Y \cup \left\{ {y}\right\}$

By Subset Relation is Transitive:

- $Y \subseteq P_y$

By Subset Relation is Transitive:

- $I \subseteq P_y$

By definition of singleton:

- $y \in \left\{ {y}\right\}$

By definition of union:

- $y \in Y \cup \left\{ {y}\right\}$

By definition of subset:

- $y \in P_y$

We will prove that

- $P_y \cap F \ne \varnothing$

Aiming for a contradiction suppose that

- $P_y \cap F = \varnothing$

By definition of $X$:

- $P_y \in X$

By definition of minimal set:

- $Y = P_y$

A contradiction between $y \notin Y$ and $y \in P_y$

$\Box$

By definitions of non-empty set and intersection:

- $\exists v: v \in P_y \land v \in F$

Define $P_x = \operatorname{finsups}\left({Y \cap \left\{ {x}\right\} }\right)^\preceq$

Analogically:

- $\exists u: u \in P_x \land u \in F$

By Finite Subset Bounds Element of Finite Suprema Set and Lower Closure:

- $\exists u' \in Y: u \preceq u' \vee \sup \left\{ {x}\right\}$

By Finite Subset Bounds Element of Finite Suprema Set and Lower Closure:

- $\exists v' \in Y: v \preceq v' \vee \sup \left\{ {y}\right\}$

- $\left({v' \vee u'}\right) \vee x = v' \vee \left({u' \vee x}\right)$

- $\left({v' \vee u'}\right) \vee x \succeq u' \vee x$

- $\sup \left\{ {x}\right\} = x$

By definition of transitivity:

- $u \preceq v' \vee u' \vee x$

By definition of upper set:

- $u' \vee v' \vee x \in F$

Analogically:

- $u' \vee v' \vee y \in F$

By Filtered in Meet Semilattice:

- $\left({u' \vee v' \vee x}\right) \wedge \left({u' \vee v' \vee y}\right) \in F$

By definition of distributive lattice:

- $\left({u' \vee v'}\right) \vee \left({x \wedge y}\right) \in F$

By Directed in Join Semilattice:

- $u' \vee v' \in Y$

By Directed in Join Semilattice:

- $\left({u' \vee v'}\right) \vee \left({x \wedge y}\right) \in Y$

This contradicts $Y \cap F = \varnothing$

$\blacksquare$

## Sources

- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott:
*A Compendium of Continuous Lattices*

- Mizar article WAYBEL_7:23