Mapping Assigning to Element Its Compact Closure Preserves Infima and Directed Suprema

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Theorem

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

Let $C = \struct {\map K L, \preceq'}$ be an ordered subset of $L$

where $\map K L$ denotes the compact subset of $L$.

Let $P = \struct {\powerset {\map K L}, \precsim}$ be an inclusion ordered set of power set of $\map K L$.


Then there exists $f: S \to \powerset {\map K L}$ such that $f$ preserves infima and directed suprema and is an injection and $\forall x \in S: \map f x = x^{\mathrm{compact} }$

where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Proof

By definitions of compact subset, compact closure, and subset:

$\forall x \in S: x^{\mathrm{compact} } \subseteq \map K L$

By definition of power set:

$\forall x \in S: x^{\mathrm{compact} } \in \powerset {\map K L}$

Define a mapping $f:S \to \powerset {\map K L}$ such that:

$\forall x \in S: \map f x = x^{\mathrm{compact} }$

By Compact Closure is Directed:

$\forall x \in S: x^{\mathrm{compact} }$ is directed.

By definition of ordered subset:

$\forall x \in S: x^{\mathrm{compact} }$ is directed in $C$.

We will prove that

$\forall x \in S: x^{\mathrm{compact} }$ is a lower section in $C$.

Let $x \in S$.

Let $y \in x^{\mathrm{compact} }, z \in \map K L$ such that:

$z \preceq y$

By definition of compact closure:

$y \preceq x$

By definition of transitivity:

$z \preceq x$

By definition of compact subset:

$z$ is compact.

Thus by definition of compact closure:

$z \in x^{\mathrm{compact} }$

$\Box$


By Bottom in Compact Closure:

$\forall x \in S: \bot_L \in x^{\mathrm{compact} }$

where $\bot_L$ denotes the bottom in $L$.

By definition:

$\forall x \in S: x^{\mathrm{compact} }$ is non-empty.

By definition of ideal in ordered set:

$\forall x \in S: x^{\mathrm{compact} } \in \map {\operatorname{Ids} } C$

where $\map {\operatorname{Ids} } C$ denotes the set of all ideals in $C$.

Then

$f: S \to \map {\operatorname{Ids} } C$

Define $I = \struct {\map {\operatorname{Ids} } C, \precsim'}$ as an inclusion ordered set.

By definitions of $\operatorname{Ids}$ and power set:

$\map {\operatorname{Ids} } C \subseteq \powerset {\map K L}$

By Mapping Assigning to Element Its Compact Closure is Order Isomorphism:

$f$ is order isomorphism between $L$ and $I$.

By Bottom in Compact Subset and Compact Subset is Join Subsemilattice:

$C$ is bounded below join semilattice.

By Ideals are Continuous Lattice Subframe of Power Set:

$I$ is a continuous lattice subframe of $P$.

By Power Set is Complete Lattice:

$P$ is a complete lattice.

By Order Isomorphism Preserves Infima and Suprema:

$f: S \to \map {\operatorname{Ids} } C$ preserves infima

and

$f: S \to \map {\operatorname{Ids} } C$ preserves suprema.

Thus by Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima:

$f: S \to \powerset {\map K L}$ preserves infima.

By definition:

$f: S \to \map {\operatorname{Ids} } C$ preserves directed suprema.

Thus by Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema:

$f: S \to \powerset {\map K L}$ preserves directed suprema.

By definition of order isomorphism:

$f: S \to \map {\operatorname{Ids} } C$ is a bijection.

By definition of bijection:

$f: S \to \map {\operatorname{Ids} } C$ is an injection.

Thus by definition of injection:

$f: S \to \powerset {\map K L}$ is an injection.

Thus

$\forall x \in S: \map f x = x^{\mathrm{compact} }$

$\blacksquare$


Sources

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