algebraic lattice




An algebraic lattice is a lattice which is

An algebraic lattice is a complete lattice (equivalently, a suplattice, or in different words a poset with the property of having arbitrary colimits but with the structure of directed colimits/directed joins) in which every element is the supremum of the compact elements below it (an element ee is compact if, for every subset SS of the lattice, ee is less than or equal to the supremum of SS just in case ee is less than or equal to the supremum of some finite subset of SS).

Here is an alternative formulation:


An algebraic lattice is a poset which is locally finitely presentable as a category.

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 SetSet).

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 TT, the subobject lattice of an object in TT-AlgAlg 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 2 below).

It is trivial that every finite lattice is algebraic.


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-continuous functions between them; i.e., those functions which preserve directed joins (hence the parenthetical remarks above).

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

There is a full embedding

i:AlgLatTop 0i \colon AlgLat \to Top_0

to the category of T 0T_0-spaces, taking an algebraic lattice LL to the space whose points are elements of LL, and whose open sets UU are defined by the property that their characteristic maps

χ U:L2\chi_U: L \to \mathbf{2}

(χ U(a)=1\chi_U(a) = 1 if aUa \in U, else χ U(a)=0\chi_U(a) = 0) are poset maps that preserve directed colimits. The specialization order of i(L)i(L) is LL again.

Every T 0T_0-space XX occurs as a subspace of some space i(L)i(L) associated with an algebraic lattice. Explicitly, let L(X)L(X) be the power set of the underlying set of the topology, P|𝒪(X)|P{|\mathcal{O}(X)|}, and define

X(iL)(X)X \to (i\circ L)(X)

to take xx to N(x){U𝒪(X):xU}N(x) \coloneqq \{U \in \mathcal{O}(X): x \in U\}. This gives a topological embedding of XX in i(L(X))i(L(X)).


On similar grounds, if U:AlgLatSetU \colon AlgLat \to Set is the forgetful functor, then the 2-image of the projection functor π:SetUSet\pi \colon Set\downarrow U \to Set is the category of topological spaces TopTop. In more nuts-and-bolts terms, an object (S,L,f:SU(L))(S, L, f \colon S \to U(L)) gives a space with underlying set SS and open sets those of the form f 1(O)f^{-1}(O), where OO ranges over the Scott topology on LL. Notice that if (f:SS,g:LL)(f \colon S \to S', g \colon L \to L') is a morphism in SetUSet \downarrow U, then ff is continuous with respect to these topologies. Therefore the projection π:SetUSet\pi \colon Set \downarrow U \to Set factors through the faithful forgetful functor TopSetTop \to Set. Thus, working in the factorization system (eso+full, faithful) on CatCat, we have a faithful functor 22-im(π)Topim(\pi) \to Top filling in as the diagonal

SetU Top 2-im(π) Set.\array{ Set \downarrow U & \to & Top \\ \downarrow & \nearrow & \downarrow \\ 2\text{-}im(\pi) & \to & Set. }

But notice also that SetUTopSet \downarrow U \to Top is eso and full. It is eso because any topology 𝒪(S)\mathcal{O}(S) on SS can be reconstituted from the triple (S,P|𝒪(S)|,xN(x):SP|𝒪(S)|)(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 XXX \to X' between topological spaces induces a continuous map between their T 0T_0 reflections X 0X 0X_0 \to X_{0}', and since algebraic lattices like P|𝒪(X)|P{|\mathcal{O}(X)|} (being continuous lattices) are injective objects in the category of T 0T_0 spaces, we are able to complete to a diagram

X X 0 P|𝒪(X)| X X 0 P|𝒪(X)|\array{ X & \to & X_0 & \to & P{|\mathcal{O}(X)|} \\ \downarrow & & \downarrow & & \downarrow \\ X' & \to & X_{0}' & \to & P{|\mathcal{O}(X')|} }

where the rightmost vertical arrow is Scott-continuous (and the horizontal composites are of the form xN(x)x \mapsto N(x)). Finally, since SetUTopSet \downarrow U \to Top is eso and full, it follows that 22-im(π)Topim(\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) partial equivalence relations over AlgLatAlgLat, or equivalently of (set-theoretic) total equivalence relations on T 0T_0 topological spaces.

Relation to locally finitely presentable categories

One of our definitions of algebraic lattice is: a poset LL which is locally finitely presentable when viewed as a category. The completeness of LL means that right adjoints LSetL \to Set are representable, given by L(p,):LSetL(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 pp, which in lattice theory are usually called compact elements. These compact elements are closed under finite joins.

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



If CC is a locally finitely presentable category and XX is an object of CC, then

  • The lattice of subobjects Sub(X)Sub(X),

  • the lattice of quotient objects (equivalence classes of epis sourced at XX) Quot(X)Quot(X),

  • the lattice of congruences (internal equivalence relations) on XX

are all algebraic lattices.

This is due to Porst. Of course if CC 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 CC is an exact category.

Congruence lattices

The following result is due to Grätzer and Schmidt:


Every algebraic lattice is isomorphic to the congruence lattice of some model XX of some finitary algebraic theory.

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 finite algebra XX. It has been conjectured that this is in fact false: see this MO discussion.

Another problem which had long remained open is the congruence lattice problem: is every distributive algebraic lattice the congruence lattice (or lattice of quotient objects) of some lattice LL? The answer is negative, as shown by Wehrung in 2007: see this Wikipedia article.

Completely distributive lattices


The category of Alexandroff locales is equivalent to that of completely distributive algebraic lattices.

This appears as (Caramello, remark 4.3).

The completely distributive algebraic lattices form a reflective subcategory of that of all distributive lattices. The reflector is called canonical extension.

See also compact element, compact element in a locale?.

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

(n,r)-categoriestoposeslocally presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický’s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger’s theoremglobal model structures on simplicial presheavesn/a
(∞,1)-topos theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories


The relation to locally finitely presentable categories is discussed in

  • Hans Porst, Algebraic lattices and locally finitely presentable categories (pdf)

That every algebraic lattice is a congruence lattice is proved in

  • G. Grätzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59.

Revised on March 5, 2015 16:06:46 by Noam Zeilberger (