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\begin{document}

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\section*{algebraic lattice}

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

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Def}{Definition}\dotfill \pageref*{Def} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_category_of_algebraic_lattices}{The category of algebraic lattices}\dotfill \pageref*{the_category_of_algebraic_lattices} \linebreak
\noindent\hyperlink{RelationToLocallyFinitelyPresentableCategories}{Relation to locally finitely presentable categories}\dotfill \pageref*{RelationToLocallyFinitelyPresentableCategories} \linebreak
\noindent\hyperlink{congruence_lattices}{Congruence lattices}\dotfill \pageref*{congruence_lattices} \linebreak
\noindent\hyperlink{completely_distributive_lattices}{Completely distributive lattices}\dotfill \pageref*{completely_distributive_lattices} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Def}{}\subsection*{{Definition}}\label{Def}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{algebraic lattice} is a [[lattice]] which is

\begin{itemize}%
\item a [[complete lattice]];


\item such that every element is a [[join]] of [[compact elements]].



\end{itemize}
\end{defn}
An \textbf{algebraic lattice} is a [[complete lattice]] (equivalently, a [[suplattice]], or in different words a [[poset]] with the [[extra property|property]] of having arbitrary [[colimits]] but with the [[structure]] of [[directed colimits]]/[[directed joins]]) in which every element is the [[supremum]] of the [[compact element]]s below it (an element $e$ is compact if, for every subset $S$ of the lattice, $e$ is less than or equal to the supremum of $S$ just in case $e$ is less than or equal to the supremum of some finite subset of $S$).

Here is an alternative formulation:

\begin{defn}
\label{}\hypertarget{}{}
An algebraic lattice is a [[poset]] which is [[locally finitely presentable category|locally finitely presentable]] as a category.

\end{defn}
This formulation suggests a useful way of viewing algebraic lattices in terms of [[Gabriel-Ulmer duality]] (but with regard to enrichment in [[truth values]], instead of in $Set$).

As this last formulation suggests, algebraic lattices typically arise as [[subobject lattices]] for objects in locally finitely presentable categories. As an example, for any (finitary) [[Lawvere theory]] $T$, the subobject lattice of an object in $T$-$Alg$ is an algebraic lattice (this class of examples explains the origin of the term ``algebraic lattice'', which is due to Garrett Birkhoff). In fact, all algebraic lattices arise this way (see Theorem \ref{GS} below).

It is trivial that every finite lattice is algebraic.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{the_category_of_algebraic_lattices}{}\subsubsection*{{The category of algebraic lattices}}\label{the_category_of_algebraic_lattices}

The [[morphisms]] most commonly considered between algebraic lattices are the [[finitary functors]] between them, which is to say, the [[Scott topology|Scott-continuous]] functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks \hyperlink{Def}{above}).

The resulting category \textbf{AlgLat} is [[cartesian closed]] and is dually equivalent to the category whose objects are [[meet semilattices]] (construed as categories with [[finite limits]] [[enriched category|enriched]] over [[truth values]]) and whose morphisms are meet-preserving [[profunctors]] between them (using the convention that a $V$-enriched profunctor from $C$ to $D$ is a functor $D^{op} \times C \rightarrow V$; of course, with an opposite convention, one could similarly state a covariant equivalence).

There is a \emph{full} [[full embedding|embedding]]

\begin{displaymath}
i \colon AlgLat \to Top_0
\end{displaymath}
to the category of $T_0$-[[separation axioms|spaces]], taking an algebraic lattice $L$ to the space whose points are elements of $L$, and whose [[open sets]] $U$ are defined by the property that their [[characteristic maps]]

\begin{displaymath}
\chi_U: L \to \mathbf{2}
\end{displaymath}
($\chi_U(a) = 1$ if $a \in U$, else $\chi_U(a) = 0$) are poset maps that preserve [[directed colimits]]. The [[specialization order]] of $i(L)$ is $L$ again.

Every $T_0$-space $X$ occurs as a [[subspace]] of some space $i(L)$ associated with an algebraic lattice. Explicitly, let $L(X)$ be the [[power set]] of the underlying set of the [[topology]], $P{|\mathcal{O}(X)|}$, and define

\begin{displaymath}
X \to (i\circ L)(X)
\end{displaymath}
to take $x$ to $N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}$. This gives a topological embedding of $X$ in $i(L(X))$.

\begin{uremark}
On similar grounds, if $U \colon AlgLat \to Set$ is the forgetful functor, then the \href{http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property#a_factorisation_system_14}{2-image} of the projection functor $\pi \colon Set\downarrow U \to Set$ is the category of topological spaces $Top$. In more nuts-and-bolts terms, an object $(S, L, f \colon S \to U(L))$ gives a space with underlying set $S$ and open sets those of the form $f^{-1}(O)$, where $O$ ranges over the Scott topology on $L$. Notice that if $(f \colon S \to S', g \colon L \to L')$ is a morphism in $Set \downarrow U$, then $f$ is continuous with respect to these topologies. Therefore the projection $\pi \colon Set \downarrow U \to Set$ factors through the faithful forgetful functor $Top \to Set$. Thus, working in the factorization system (eso+full, faithful) on $Cat$, we have a faithful functor $2$-$im(\pi) \to Top$ filling in as the diagonal

\begin{displaymath}
\itexarray{
Set \downarrow U & \to & Top \\
\downarrow & \nearrow & \downarrow \\
2\text{-}im(\pi) & \to & Set.
}
\end{displaymath}
But notice also that $Set \downarrow U \to Top$ is \href{http://ncatlab.org/nlab/show/ternary+factorization+system#examples_9}{eso and full}. It is eso because any topology $\mathcal{O}(S)$ on $S$ can be reconstituted from the triple $(S, P{|\mathcal{O}(S)|}, x \mapsto N(x) \colon S \to P{|\mathcal{O}(S)|})$. We claim it is full as well. For, every continuous map $X \to X'$ between topological spaces induces a continuous map between their $T_0$ reflections $X_0 \to X_{0}'$, and since algebraic lattices like $P{|\mathcal{O}(X)|}$ (being continuous lattices) are [[injective objects]] in the category of $T_0$ spaces, we are able to complete to a diagram

\begin{displaymath}
\itexarray{
X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\
\downarrow & & \downarrow & & \downarrow \\
X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|}
}
\end{displaymath}
where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form $x \mapsto N(x)$). Finally, since $Set \downarrow U \to Top$ is eso and full, it follows that $2$-$im(\pi) \to Top$ is eso, full, and faithful, and therefore an equivalence of categories.

This connection is explored in more depth with the category of [[equilogical spaces]], which can be seen either as a category of (set-theoretic) [[equivalence relation|partial equivalence relations]] over $AlgLat$, or equivalently of (set-theoretic) total [[equivalence relations]] on $T_0$ topological spaces.

\end{uremark}
\hypertarget{RelationToLocallyFinitelyPresentableCategories}{}\subsubsection*{{Relation to locally finitely presentable categories}}\label{RelationToLocallyFinitelyPresentableCategories}

One of our definitions of algebraic lattice is: a poset $L$ which is locally finitely presentable when viewed as a category. The completeness of $L$ means that right adjoints $L \to Set$ are representable, given by $L(p, -) \colon L \to Set$, and we are particularly interested in those representable functors that preserve [[filtered colimits]]. These correspond precisely to finitely presentable objects $p$, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.

By [[Gabriel-Ulmer duality]], $L$ is determined from the join-semilattice of compact elements $K$ by $L \cong Lex(K^{op}, Set)$. Since the elements of $K^{op}$ are subterminal, we can also write $L \cong Lex(K^{op}, 2)$ where $2 = Sub(1)$.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Porst)}

If $C$ is a [[locally finitely presentable category]] and $X$ is an object of $C$, then

\begin{itemize}%
\item The lattice of subobjects $Sub(X)$,


\item the lattice of quotient objects (equivalence classes of epis sourced at $X$) $Quot(X)$,


\item the lattice of congruences (internal equivalence relations) on $X$



\end{itemize}
are all algebraic lattices.

\end{theorem}
This is due to \hyperlink{Porst}{Porst}. Of course if $C$ is the category of algebras of an Lawvere theory, then the lattice of quotient objects of an algebra is isomorphic to its congruence lattice, as such $C$ is an [[exact category]].

\hypertarget{congruence_lattices}{}\subsubsection*{{Congruence lattices}}\label{congruence_lattices}

The following result is due to Gr\"a{}tzer and Schmidt:

\begin{theorem}
\label{GS}\hypertarget{GS}{}
Every algebraic lattice is isomorphic to the congruence lattice of some [[model]] $X$ of some finitary algebraic theory.

\end{theorem}
In particular, since every finite lattice is algebraic, every finite lattice arises this way. Remarkably, it is not known at this time whether every finite lattice arises as the congruence lattice of a \emph{finite} algebra $X$. It has been conjectured that this is in fact \textbf{false}: see this \href{http://mathoverflow.net/a/196074/2926}{MO discussion}.

Another problem which had long remained open is the congruence lattice problem: is every \emph{distributive} algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice $L$? The answer is negative, as shown by Wehrung in 2007: see this \href{http://en.m.wikipedia.org/wiki/Congruence_lattice_problem}{Wikipedia article}.

\hypertarget{completely_distributive_lattices}{}\subsubsection*{{Completely distributive lattices}}\label{completely_distributive_lattices}

\begin{prop}
\label{}\hypertarget{}{}
The category of [[Alexandroff locales]] is equivalent to that of [[completely distributive lattice|completely distributive]] algebraic lattices.

\end{prop}
This appears as (\hyperlink{Caramello}{Caramello, remark 4.3}).

The [[completely distributive lattice|completely distributive]] algebraic lattices form a [[reflective subcategory]] of that of all distributive lattices. The reflector is called \emph{[[canonical extension]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

See also [[compact element]], [[compact element in a locale]].

[[!include locally presentable categories - table]]

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

\begin{itemize}%
\item [[Andrej Bauer]], [[Lars Birkedal]], [[Dana Scott]], \emph{Equilogical Spaces}, Theoretical Computer Science, 315(1):35-59, 2004. (\href{http://math.andrej.com/2002/07/05/equilogical-spaces/}{web})


\item Olivia Caramello, \emph{A topos-theoretic approach to Stone-type dualities} (\href{http://arxiv.org/abs/1103.3493}{arXiv:1103.3493})



\end{itemize}
The relation to [[locally finitely presentable categories]] is discussed in

\begin{itemize}%
\item [[Hans Porst]], \emph{Algebraic lattices and locally finitely presentable categories} (\href{http://www.math.uni-bremen.de/~porst/dvis/PORST_AlgebraicLattices_revfinAU.pdf}{pdf})

\end{itemize}
That every algebraic lattice is a congruence lattice is proved in

\begin{itemize}%
\item G. Gr\"a{}tzer and E. T. Schmidt, \emph{Characterizations of congruence lattices of abstract algebras}, Acta Sci. Math. (Szeged) 24 (1963), 34--59.

\end{itemize}
[[!redirects algebraic lattice]] [[!redirects algebraic lattices]]



\end{document}