Ideals form Arithmetic Lattice

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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 $z = x \cap y$.

By Meet in Set of Ideals:

$z = x \wedge_I y$

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:

$c \preceq a \wedge b$

By Meet Precedes Operands:

$a \wedge b \preceq a$ 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:

$I$ is an algebraic lattice.

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

$I$ is an arithmetic lattice.

$\blacksquare$


Sources