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\begin{document}

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\section*{Moore closure}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{moore_closures}{}\section*{{Moore closures}}\label{moore_closures}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{InTermsOfClosureCondition}{In terms of closure condition}\dotfill \pageref*{InTermsOfClosureCondition} \linebreak
\noindent\hyperlink{InTermsOfClosureOperators}{In terms of closure operators}\dotfill \pageref*{InTermsOfClosureOperators} \linebreak
\noindent\hyperlink{InTermsOfMonads}{In terms of monads}\dotfill \pageref*{InTermsOfMonads} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{generalisations}{Generalisations}\dotfill \pageref*{generalisations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{Moore closure} is a very general idea of what it can mean for a [[set]] to be \emph{closed} under some condition. It includes, as special cases, the operation of [[closed subspace|closure]] in a [[topological space]], many examples of generation of structures from [[base|bases]] and even [[subbase|subbases]], and generating [[subalgebras]] from subsets of an algebra.

Secretly, it is the same thing as the collection of subsets preserved by some [[monad]] on a [[power set]] (the subset of ``[[modal types]]''). In fact it is a special case of the notion of [[closure operator]] or [[modality]] in [[logic]]/[[type theory]], namely the special case where the ambient [[category]]/[[hyperdoctrine]] is the [[topos]] [[Set]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

We give two equivalent definitions. The first one

\begin{itemize}%
\item \hyperlink{InTermsOfClosureCondition}{In terms of closure condition}

\end{itemize}
gives the explicit condition for a [[subset]] of a [[power set]] to qualify as a Moore closure, the second

\begin{itemize}%
\item \hyperlink{InTermsOfClosureOperators}{In terms of closure operators}

\end{itemize}
characterizes Moore closures as the collections of [[modal types]] of suitable [[closure operators]]. More abstractly, this characterizes Moore closures

\begin{itemize}%
\item \hyperlink{InTermsOfMonads}{In terms of monads}

\end{itemize}
on the [[subobject lattice]] of the given set.

\hypertarget{InTermsOfClosureCondition}{}\subsubsection*{{In terms of closure condition}}\label{InTermsOfClosureCondition}

\begin{defn}
\label{}\hypertarget{}{}
Let $X$ be a [[set]], and let $\mathcal{C} \subset P X$ be a collection of [[subsets]] of $X$. Then $\mathcal{C}$ is a \textbf{Moore collection} if every [[intersection]] of members of $\mathcal{C}$ belongs to $\mathcal{C}$.

That is, given a family $(A_i)_i$ of sets in $X$,

\begin{displaymath}
\forall i,\; A_i \in \mathcal{C} \;\Rightarrow\; \bigcap_i A_i \in \mathcal{C} .
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Given any collection $\mathcal{B}$ whatsoever of subsets of $X$, the Moore collection \textbf{generated} by $\mathcal{B}$ is the collection of all intersections of members of $\mathcal{B}$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is indeed a Moore collection, and it equals $\mathcal{B}$ if and only if $\mathcal{B}$ is a Moore collection.

\end{remark}
\hypertarget{InTermsOfClosureOperators}{}\subsubsection*{{In terms of closure operators}}\label{InTermsOfClosureOperators}

\begin{defn}
\label{ClosureOperator}\hypertarget{ClosureOperator}{}
Again let $X$ be a set, and now let $Cl$ be an operation on subsets of $X$. Then $Cl$ is a \textbf{closure operation} if $Cl$ is monotone, isotone, and idempotent. That is,

\begin{enumerate}%
\item $A \subseteq B \;\Rightarrow\; Cl(A) \subseteq Cl(B)$,
\item $A \subseteq Cl(A)$, and
\item $Cl(Cl(A)) \subseteq Cl(A)$ (the reverse inclusion follows from the previous two properties).

\end{enumerate}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
If $Cl$ is a closure operation, then let $\mathcal{C}$ be the collection of sets that equal their own closures (the ``[[modal types]]'' or ``[[local objects]]''). Then $\mathcal{C}$ is a Moore collection.

Conversely, if $\mathcal{C}$ is a Moore collection, then let $Cl(A)$ be the intersection of all closed sets that contain $A$. Then $Cl$ is a closure operator.

Furthermore, the two maps above, from closure operators to Moore collections and vice versa, are inverses.

\end{prop}
\hypertarget{InTermsOfMonads}{}\subsubsection*{{In terms of monads}}\label{InTermsOfMonads}

Moore closures on $X$ are precisely [[monads]] on the [[subobject lattice]] $\mathcal{P}X$. The property (1) of a closure operator, def. \ref{ClosureOperator} ,corresponds the action of the monad on morphisms, while (2,3) are the [[unit of an adjunction|unit]] and multiplication of the monad. (The rest of the requirements of a monad are trivial in a [[poset]], since they state the equality of various morphisms with common source and target.)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

What are examples? Better to ask what \emph{isn't} an example! (Answer: preclosure in a [[pretopological space]], even though some authors call this `closure'.)

Of course, the [[closed subsets]] in a [[topological space]] form a Moore collection; then the closure of a set $A$ is its closure in the usual sense. In fact, a topological space can be \emph{defined} as a set equipped with a Moore closure with either of these additional properties (which are equivalent):

\begin{itemize}%
\item $Cl(\empty) = \empty$ and $Cl(A \cup B) = Cl(A) \cup Cl(B)$.
\item $\empty$ is closed, and so is $A \cup B$ if $A$ and $B$ are closed.

\end{itemize}
(However, these properties may fail in [[constructive mathematics]]; in fact, a topology cannot be constructively recovered from its closure operation.)

Here are some algebraic examples:

\begin{itemize}%
\item The [[subgroup]]s of a [[group]] $G$ form a Moore collection; the closure of a subset $B$ of $G$ is the subgroup generated by $B$.


\item The subrings of a [[ring]] $R$ form a Moore collection; the closure of a subset $B$ of $R$ is the subring generated by $B$.


\item The subspaces of a [[vector space]] $V$ form a Moore collection; the closure of a subset $B$ of $V$ is the subspace spanned by $B$.


\item OK, you get the idea. This applies to any [[algebraic theory]]. For a [[Lawvere theory|finitary algebraic theory]], the lattice of closed elements is an [[algebraic lattice]].



\end{itemize}
But also:

\begin{itemize}%
\item The [[normal subgroup]]s of $G$ form a Moore collection; the closure of $B$ is the normal subgroup generated by $B$.


\item The [[ideal]]s of a ring $R$ form a Moore collection; the closure of $B$ is the ideal generated by $B$.


\item The (topologically) closed subspaces of a [[Hilbert space]] $H$ form a Moore collection; the closure of $B$ is the closed subspace generated by $B$.


\item And many further examples.



\end{itemize}
Here are some examples on [[power set]]s:

\begin{itemize}%
\item The topologies on $X$ form a Moore collection on $\mathcal{P}X$; the closure of a subset $\mathcal{B}$ of $\mathcal{P}X$ is the topology generated by $\mathcal{B}$ as a [[subbase]].


\item The [[filter]]s on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the filter generated by $\mathcal{B}$ as a [[subbase]]. (The \emph{proper} filters on $X$ do \emph{not} form a Moore collection; not every $\mathcal{B}$ generates a proper filter.)


\item The $\sigma$-[[sigma-algebra|algebras]] on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the $\sigma$-algebra generated by $\mathcal{B}$. (This is the `abstract nonsense' way to generate a $\sigma$-algebra; else you have to do transfinite induction on countable [[ordinal number|ordinals]].)


\item And so on.



\end{itemize}
Topping off these, the Moore collections on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the Moore collection generated by $\mathcal{B}$ as described in the definitions.

See also at \emph{[[matroid]]}.

\hypertarget{generalisations}{}\subsection*{{Generalisations}}\label{generalisations}

The definition of Moore collection really makes sense in any [[inflattice]]; even better, the definition of closure operator makes sense in any [[partial order|poset]]. This context is the generic meaning of [[closure operator]]; here are some examples:

\begin{itemize}%
\item Instead of $\mathcal{P}X$, work in the [[opposite category|opposite poset]] $\mathcal{P}^{op}X$. Then the open sets in a [[topological space]] $X$ form a Moore collection whose closure operator is the usual interior operation. Now we can define a topological space as a set equipped with a Moore closure operator on $\mathcal{P}^{op}X$ that preserves [[join]]s (which here are [[intersection]]s); this definition is even valid constructively.


\item Let $f \dashv g$ be a [[Galois connection]] between [[partial order|posets]] $A$ and $B$. Call an element of $A$ \emph{normal} if $g(f(a)) \leq a$ (the reverse is always true). Then $g \circ f$ is a closure operator. This generalises the case of the normal subgroups of $G$ when $G$ is the [[Galois group]] of an extension of [[field]]s.



\end{itemize}
Since Galois connections are simply [[adjunction]]s between posets, the concept of Moore closure cries out for [[categorification]]. And in fact, the answer is well known in category theory: it is a [[monad]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

Section 4.1--4.12 in

\begin{itemize}%
\item Erich Schechter, \emph{[[Handbook of Analysis and its Foundations]]}

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Closure_operator}{Closure operator}}

\end{itemize}
[[!redirects Moore closure]] [[!redirects Moore closures]] [[!redirects Moore clsoure]] [[!redirects Moore collection]] [[!redirects Moore collections]]



\end{document}