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

== 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:
:$z \in x^{\mathrm{compact} }$
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
:$\forall x \in S: \map f x = x^{\mathrm{compact} }$
== Sources ==
* 
* 
Category:Continuous Lattices
Category:Way Below Relation