nLab
filter

Filters

Idea

A filter is dual to an ideal.

Definitions

A subset FF of a poset LL is called a filter if it is upward-closed and downward-directed; that is:

  1. If ABA \leq B in LL and AFA \in F, then BFB \in F;
  2. for some AA in LL, AFA \in F;
  3. if AFA \in F and BFB \in F, then for some CFC \in F, CAC \leq A and CBC \leq B.

Sometimes the term ‘filter’ is used for an upper set, that is any set satisfying axiom (1). (Ultimately this connects with the use of ‘ideal’ in monoid theory.)

In a lattice, one can use these alternative axioms:

  1. If AFA \in F and BB in LL, then ABFA \vee B \in F;
  2. F\top \in F;
  3. if AFA \in F and BFB \in F, then ABFA \wedge B \in F.

Here, (1) is equivalent to the previous version; the others, which here say that the lattice is closed under finite meets, are equivalent given (1). (These axioms look more like the axioms for an ideal of a ring.)

You can also interpret these axioms to say that, if you think of FF as a function from LL to the set TVTV of truth values, then FF is a homomorphism of meet-semilattices.

A filter of subsets of a given set SS is a filter in the power set of SS. One also sees filters of open subsets, filters of compact subsets, etc, especially in topology.

Kinds of filters

A filter FF is proper if there exists an element AA of LL such that AFA \notin F. A filter in a lattice is proper iff F\bot \notin F; in particular, a filter of subsets of SS is proper iff F\empty \notin F. In constructive mathematics, however, one usually wants a stronger (but classically equivalent) notion: a filter FF of subsets of SS is proper if every element of FF is inhabited. If AFA \in F for every AA (in particular if F\empty \in F), then we have the improper filter. Compare proper subset and improper subset.

Filters are often assumed to be proper by default in analysis and topology, where proper filters correspond to nets. However, we will try to remember to include the adjective ‘proper’.

If the complement of a filter is an ideal, then we say that the filter is prime (and equivalently that the ideal is prime). A prime filter is necessarily proper; a proper filter in a lattice is prime iff, whenever ABFA \vee B \in F, either AFA \in F or BFB \in F. In other words, F:LTVF: L \to TV must be a homomorphism of lattices. The generalisation to arbitrary joins gives a completely prime filter.

A filter is an ultrafilter, or maximal filter, if it is maximal among the proper filters. (See that article for alternative formulations and applications.) In a distributive lattice, every ultrafilter is prime; the converse holds in a Boolean lattice. In this case, we can say that F:LTVF: L \to TV is a homomorphism of Boolean lattices.

Given an element xx of SS, the principal ultrafilter (of subsets of SS) at xx consists of every subset of SS to which xx belongs. A principal ultrafilter is also called a fixed ultrafilter; more generally, a filter of subsets is fixed if its intersection is inhabited. In contrast, if FF is an filter whose meet (of all elements) exists and is a bottom element (the empty set for a filter of subsets), then we call FF free.

Free ultrafilters on Boolean algebras are important in nonstandard analysis and model theory.

Filterbases

A subset FF of a lattice LL is a filterbase if it becomes a filter when closed under axiom (1). Equivalently, a filterbase is any downward-directed subset. Any subset of a meet-semilattice may be used as a filter subbase; form a filterbase by closing under finite meets.

A filterbase FF of sets is proper (that is, it generates a proper filter of sets) iff each set in FF is inhabited. A filter subbase of sets is proper iff it satisfies the finite intersection property (well known in topology from a criterion for compact spaces): every finite collection from the subfilter has inhabited intersection.

The poset of filters and push-forwards

If f:L 1L 2f:L_1\to L_2 is a monotone? map and FL 1F\subseteq L_1 a filter, then {f(A)AF}L 2\lbrace f(A) \mid A\in F\rbrace\subseteq L_2 is a filter base. Let f *(F)f_\ast(F) be the filter generated by this filter base. The set of filters Filters(L)Filters(L) is a poset in its own right w.r.t. inclusion and f *:(Filters(L 1),)(Filters(L 2),)f_\ast: (Filters(L_1),\subseteq)\to(Filters(L_2),\subseteq) is monotone. Therefore LFilters(L),ff *L\mapsto Filters(L), f\mapsto f_\ast is a functor from the category of posets to itself.

If ff satisfies stronger properties than mere monotony, then f *f_\ast will be better behaved as well:

  • If LL is a meet-semilattice, then Filters(L)Filters(L) is a complete join-semilattice: iIF i= jJA jJI finite,A jF j\bigvee_{i\in I} F_i = \left\langle \bigwedge_{j\in J} A_j \mid J\subseteq I\text{ finite}, A_j\in F_j\right\rangle. If f:L 1L 2f:L_1\to L_2 is a monotone map between meet-semilattices (i.e. f(AA)f(A)f(A)f(A\wedge A')\leq f(A)\wedge f(A') holds, but equality need not hold!), then f *f_\ast automatically respects all joins.

  • If LL has all |I||I|-fold joins for some indexing set II, then Filters(L)Filters(L) has |I||I|-fold meets: iIF i={ iIA iA iF i}\bigwedge_{i\in I} F_i = \lbrace \bigvee_{i\in I} A_i \mid A_i\in F_i\rbrace. If ff respects |I||I|-fold joins, then f *f_\ast respects |I||I|-fold meets.

  • If LL is a distributive lattice, then Filters(L)Filters(L) is a frame, i.e. the infinite distributive law F iIG i= iI(FG i)F \wedge \bigvee_{i\in I} G_i = \bigvee_{i\in I} (F\wedge G_i) holds. The other distributive law also holds for finite meets: F iIG i= iI(FG i)F \vee \bigwedge_{i\in I} G_i = \bigwedge_{i\in I} (F\vee G_i) if II is finite. It holds more generally if LL has all |I||I|-fold meets, the |I||I|-fold distributive law A iI= iI(AB i)A\wedge \bigvee_{i\in I} = \bigvee_{i\in I} (A\wedge B_i) holds in LL and FF is closed under |I||I|-fold meets.

  • If LL is a frame, then Filters(L)Filters(L) is a complete lattice. If ff respects all joins, then f *f_\ast respects all joins and all meets. In this case f *f_\ast has both a right and a left adjoint f !,f *:L 2L 1f^!, f^\ast: L_2\to L_1 which are given by f *(G)={F|Gf *(F)}f^\ast(G) = \bigwedge \lbrace F | G\subseteq f_\ast(F)\rbrace and f !(G)={F|f *(F)G}f^!(G) = \bigvee \lbrace F | f_\ast(F)\subseteq G\rbrace respectively so that f *(G)FGf *(F)f^\ast(G)\subseteq F \iff G\subseteq f_\ast(F) and f *(F)HFf !(H)f_\ast(F)\subseteq H \iff F\subseteq f^!(H) hold.

Examples

  • If L 1=(P(X),),L 2=(P(Y),)L_1=(P(X),\subseteq), L_2=(P(Y),\subseteq) for some sets X,YX,Y and α:XY\alpha:X\to Y is any map, then f:L 1L 2,Aα(A)f:L_1\to L_2, A\mapsto \alpha(A) is monotone and respects arbitrary joins. Therefore f *:Filters(L 1)Filters(L 2)f_\ast: Filters(L_1)\to Filters(L_2) respects arbitrary meets and joins. Note that even though L 1,L 2L_1, L_2 are meet-semilattices, ff does not respect meets in general, because α(AA)α(A)α(A)\alpha(A\cap A')\subseteq \alpha(A)\cap\alpha(A') holds, but equality is equivalent to α\alpha being injective.
  • The left adjoint f *f^\ast in this case is given by taking preimage filters: f *(G)=f 1(B)|BGf^\ast(G) = \langle f^{-1}(B) | B\in G\rangle

Application to analysis and topology

Every net ν:IS\nu: I \to S defines an eventuality filter E νE_\nu: let AA belong to E νE_\nu if, for some index kk, for every lkl \geq k, ν lA\nu_l \in A. (That is, ν\nu is eventually in AA.) Note that E νE_\nu is proper; conversely, any proper filter FF has a net whose eventuality filter is FF (as described at net). Everything below can be done for nets as well as for (proper) filters, but filters often lead to a cleaner theory.

In a topological space SS, a filter FF on SS converges to a point xx of SS if every neighbourhood of xx belongs to FF. A filter FF clusters at a point xx if every neighbourhood of xx intersects every element of FF. With these definitions, the improper filter converges to every point and clusters at no point; a proper filter, however, clusters at every point that it converges to.

The concepts of continuous function and such conditions as compactness and Hausdorffness may be defined quite nicely in terms of the convergence relation. In fact, everything about topological spaces may be defined in terms of the convergence relation, although not always nicely. This is because topological spaces form a full subcategory of the category of convergence spaces, where the convergence relation is the fundamental concept. More details are there.

In a metric space SS, a filter FF on SS is Cauchy if it has elements of arbitrarily small diameter. Then a sequence is a Cauchy sequence iff its eventuality filter is Cauchy. (This can be generalised to uniform spaces.) The concept of completion of a metric space may be defined quite nicely in terms of the Cauchy filters, although not every property (not even every uniform property) of metric spaces can be defined in this way. As for convergence, there is a general notion of Cauchy space, but the forgetful functors from metric and uniform spaces are now not full.

References

Revised on January 29, 2015 20:17:55 by Johannes Hahn? (78.53.196.170)