# Ideals form Arithmetic Lattice

## Theorem

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

Let $I = \left({\mathit{Ids}\left({L}\right), \precsim}\right)$ be an inclusion ordered set

where

- $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$,
- $\mathord\precsim = \mathord\subseteq \cap \left({\mathit{Ids}\left({L}\right) \times \mathit{Ids}\left({L}\right)}\right)$

Then $I$ is an arithmetic lattice.

## Proof

We will prove that:

- $\forall x, y \in \map K I: \exists z \in \map K I: z \precsim x \land z \precsim y \land \forall v \in \map K I: v \precsim x \land v \precsim y \implies v \precsim z$

Let $x, y \in \map K I$.

By definition of compact subset:

- $x$ and $y$ are compact elements in $I$.

By Compact Element iff Principal Ideal and definition of principal ideal:

- $\exists a \in S: x = a^\preceq$

and

- $\exists b \in S: y = b^\preceq$

By Intersection of Semilattice Ideals is Ideal:

- $x \cap y \in \mathit{Ids}\left({L}\right)$

Define $z = x \cap y$.

- $z = x \wedge_I y$

We will prove that

- $\left({a \wedge b}\right)^\preceq \subseteq z$

Let $c \in \left({a \wedge b}\right)^\preceq$

By definition of lower closure of element:

- $c \preceq a \wedge b$

- $a \wedge b \preceq a$ and $a \wedge b \preceq b$

By definition of transitivity:

- $c \preceq a$ and $c \preceq b$

By definition of lower closure of element:

- $c \in x$ and $c \in y$

Thus by definition of intersection:

- $c \in z$

$\Box$

We will prove that

- $z \subseteq \left({a \wedge b}\right)^\preceq$

Let $c \in z$.

By definition of intersection:

- $c \in x$ and $c \in y$

By definition of lower closure of element:

- $c \preceq a$ and $c \preceq b$

By definitions of infimum and lower bound:

- $c \preceq a \wedge b$

Thus by definition of lower closure of element:

- $c \in \left({a \wedge b}\right)^\preceq$

$\Box$

By definition of set equality:

- $z = \left({a \wedge b}\right)^\preceq$

By definition:

- $z$ is a principal ideal.

By Compact Element iff Principal Ideal:

- $z$ is a compact element in $I$.

Thus by definition of compact subset:

- $z \in \map K I$

Thus by Meet Precedes Operands:

- $z \precsim x$ and $z \precsim y$

Let $v \in \map K I$ such that

- $v \precsim x$ and $v \precsim y$

By definition of $\precsim$:

- $v \subseteq x$ and $v \subseteq y$

By Intersection is Largest Subset:

- $v \subseteq z$

Thus by definition of $\precsim$:

- $v \precsim z$

$\Box$

By definition:

- $\map K I$ forms a meet semilattice.

By Compact Subset is Join Subsemilattice:

- $\map K I$ forms a join semilattice.

By definition:

- $\map K I$ forms a lattice.

By Ideals form Algebraic Lattice:

Thus by Arithmetic iff Compact Subset form Lattice in Algebraic Lattice:

- $I$ is an arithmetic lattice.

$\blacksquare$

## 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 WAYBEL13:14