Filter (mathematics)

In mathematics, a filter or order filter is a special subset of a partially ordered set (poset). Filters appear in order and lattice theory, but can also be found in topology, from which they originate. The dual notion of a filter is an order ideal.

The powerset lattice of the set with the upper set colored dark green. It is a filter, and even a principal filter. It is not an ultrafilter, as it can be extended to the larger nontrivial filter by including also the light green elements. Since cannot be extended any further, it is an ultrafilter.

Filters on sets were introduced by Henri Cartan in 1937[1][2] and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of this notion from sets to the more general setting of partially ordered sets. For information on order filters in the special case where the poset consists of the power set ordered by set inclusion, see the article Filter (set theory).

MotivationEdit

1. Intuitively, a filter in a partially ordered set (poset),   is a subset of   that includes as members those elements that are large enough to satisfy some given criterion. For example, if   is an element of the poset, then the set of elements that are above   is a filter, called the principal filter at   (If   and   are incomparable elements of the poset, then neither of the principal filters at   and   is contained in the other one, and conversely.)

Similarly, a filter on a set contains those subsets that are sufficiently large to contain some given thing. For example, if the set is the real line and   is one of its points, then the family of sets that include   in their interior is a filter, called the filter of neighbourhoods of   The thing in this case is slightly larger than   but it still does not contain any other specific point of the line.

The above interpretations explain conditions 1 and 3 in the section General definition: Clearly the empty set is not "large enough", and clearly the collection of "large enough" things should be "upward-closed". However, they do not really, without elaboration, explain condition 2 of the general definition. For, why should two "large enough" things contain a common "large enough" thing?

2. Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something (a point or a subset) in the space   call a filter the collection of subsets of   that might contain "what is looked for". Then this "filter" should possess the following natural structure:

  1. A locating scheme must be non-empty in order to be of any use at all.
  2. If two subsets,   and   both might contain "what is looked for", then so might their intersection. Thus the filter should be closed with respect to finite intersection.
  3. If a set   might contain "what is looked for", so does every superset of it. Thus the filter is upward-closed.

An ultrafilter can be viewed as a "perfect locating scheme" where each subset   of the space   can be used in deciding whether "what is looked for" might lie in  

From this interpretation, compactness (see the mathematical characterization below) can be viewed as the property that "no location scheme can end up with nothing", or, to put it another way, "always something will be found".

The mathematical notion of filter provides a precise language to treat these situations in a rigorous and general way, which is useful in analysis, general topology and logic.

3. A common use for a filter is to define properties that are satisfied by "almost all" elements of some topological space  [3] The entire space   definitely contains almost-all elements in it; If some   contains almost all elements of   then any superset of it definitely does; and if two subsets,   and   contain almost-all elements of   then so does their intersection. In a measure-theoretic terms, the meaning of "  contains almost-all elements of  " is that the measure of   is 0.

General definition: Filter on a partially ordered setEdit

A subset   of a partially ordered set   is an order filter or dual ideal if the following conditions hold:

  1.   is non-empty.
  2.   is downward directed: For every   there is some   such that   and  
  3.   is an upper set or upward-closed: For every   and     implies that  

  is said to be a proper filter if in addition   is not equal to the whole set   Depending on the author, the term filter is either a synonym of order filter or else it refers to a proper order filter. This article defines filter to mean order filter.

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In this case, the above definition can be characterized by the following equivalent statement: A subset   of a lattice   is a filter, if and only if it is a non-empty upper set that is closed under finite infima (or meets), that is, for all   it is also the case that  [4] A subset   of   is a filter basis if the upper set generated by   is all of   Note that every filter is its own basis.

The smallest filter that contains a given element   is a principal filter and   is a principal element in this situation. The principal filter for   is just given by the set   and is denoted by prefixing   with an upward arrow:  

The dual notion of a filter, that is, the concept obtained by reversing all   and exchanging   with   is ideal. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of maximal filters and prime filters) is to be found in the article on ideals. There is a separate article on ultrafilters.

Applying these definitions to the case where   is a vector space and   is the set of all vector subspaces of   ordered by inclusion   gives rise to the notion of linear filters and linear ultrafilters. Explicitly, a linear filter on a vector space   is a family   of vector subspaces of   such that if   and if   is a vector subspace of   that contains   then  [5] A linear filter is called proper if it does not contain   a linear ultrafilter on   is a maximal proper linear filter on  [5]

Filter on a setEdit

Definition of a filterEdit

There are two competing definitions of a "filter on a set," both of which require that a filter be a dual ideal.[6] One definition defines "filter" as a synonym of "dual ideal" while the other defines "filter" to mean a dual ideal that is also proper.

Warning: It is recommended that readers always check how "filter" is defined when reading mathematical literature.

A dual ideal[6] on a set   is a non-empty subset   of   with the following properties:

  1.   is closed under finite intersections: If   then so is their intersection.
    • This property implies that if   then   has the finite intersection property.
  2.   is upward closed/isotone:[7] If   and   then   for all subsets  
    • This property entails that   (since   is a non-empty subset of  ).

Given a set   a canonical partial ordering   can be defined on the powerset   by subset inclusion, turning   into a lattice. A "dual ideal" is just a filter with respect to this partial ordering. Note that if   then there is exactly one dual ideal on   which is  

A filter on a set may be thought of as representing a "collection of large subsets".[8]

Filter definitionsEdit

The article uses the following definition of "filter on a set."

Definition as a dual ideal: A filter on a set   is a dual ideal on   Equivalently, a filter on   is just a filter with respect the canonical partial ordering   described above.

The other definition of "filter on a set" is the original definition of a "filter" given by Henri Cartan, which required that a filter on a set be a dual ideal that does not contain the empty set:

Original/Alternative definition as a proper dual ideal: A filter[6] on a set   is a dual ideal on   with the following additional property:

  1.   is proper[9]/non-degenerate:[10] The empty set is not in   (i.e.  ).
Note: This article does not require that a filter be proper.

The only non-proper filter on   is   Much mathematical literature, especially that related to Topology, defines "filter" to mean a non-degenerate dual ideal.

Filter bases, subbases, and comparisonEdit

Filter bases and subbases

A subset   of   is called a prefilter, filter base, or filter basis if   is non-empty and the intersection of any two members of   is a superset of some member(s) of   If the empty set is not a member of   we say   is a proper filter base.

Given a filter base   the filter generated or spanned by   is defined as the minimum filter containing   It is the family of all those subsets of   which are supersets of some member(s) of   Every filter is also a filter base, so the process of passing from filter base to filter may be viewed as a sort of completion.

For every subset   of   there is a smallest (possibly non-proper) filter   containing   called the filter generated or spanned by   Similarly as for a filter spanned by a filter base, a filter spanned by a subset   is the minimum filter containing   It is constructed by taking all finite intersections of   which then form a filter base for   This filter is proper if and only if every finite intersection of elements of   is non-empty, and in that case we say that   is a filter subbase.

Finer/equivalent filter bases

If   and   are two filter bases on   one says   is finer than   (or that   is a refinement of  ) if for each   there is a   such that   For filter bases     and   if   is finer than   and   is finer than   then   is finer than   Thus the refinement relation is a preorder on the set of filter bases, and the passage from filter base to filter is an instance of passing from a preordering to the associated partial ordering.

If also   is finer than   one says that they are equivalent filter bases. If   and   are filter bases, then   is finer than   if and only if the filter spanned by   contains the filter spanned by   Therefore,   and   are equivalent filter bases if and only if they generate the same filter.

ExamplesEdit

A filter in a poset can be created using the Rasiowa–Sikorski lemma, which is often used in forcing. The set   is called a filter base of tails of the sequence of natural numbers   A filter base of tails can be made of any net   using the construction   where the filter that this filter base generates is called the net's eventuality filter. Therefore, all nets generate a filter base (and therefore a filter). Since all sequences are nets, this holds for sequences as well.

Let   be a set and   be a non-empty subset of   Then  is a filter base. The filter it generates (that is, the collection of all subsets containing  ) is called the principal filter generated by   A filter is said to be a free filter if the intersection of all of its members is empty. A proper principal filter is not free. Since the intersection of any finite number of members of a filter is also a member, no proper filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members. A nonprincipal filter on an infinite set is not necessarily free. The Fréchet filter on an infinite set   is the set of all subsets of   that have finite complement. A filter on   is free if and only if it includes the Fréchet filter. More generally, if   is a measure space for which   the collection of all   such that   forms a filter. The Fréchet filter is the case where   and   is the counting measure.

Every uniform structure on a set   is a filter on  

Filters in model theoryEdit

For every filter   on a set   the set function defined by

 
is finitely additive — a "measure" if that term is construed rather loosely. Therefore, the statement
 
can be considered somewhat analogous to the statement that   holds "almost everywhere". That interpretation of membership in a filter is used (for motivation, although it is not needed for actual proofs) in the theory of ultraproducts in model theory, a branch of mathematical logic.

Filters in topologyEdit

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces. A sequence is usually indexed by the natural numbers   which are a totally ordered set. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. If working only with certain categories of topological spaces, such as first-countable spaces for instance, sequences suffice to characterize most topological properties, but this is not true in general. However, filters (as well as nets) do always suffice to characterize most topological properties. An advantage to using filters is that they do not involve any set other than   (and its subsets) whereas sequences and nets rely on directed sets that may be unrelated to   Moreover, the set of all filters on   is a set whereas the class of all nets valued in   is not (it is a proper class).

Neighbourhood bases

Let   be the neighbourhood filter at point   in a topological space   This means that   is the set of all topological neighbourhoods of the point   It can be verified that   is a filter. A neighbourhood system is another name for a neighbourhood filter. A family   of neighbourhoods of   is a neighbourhood base at   if   generates the filter   This means that each subset   of   is a neighbourhood of   if and only if there exists   such that  

Convergent filters and cluster points

We say that a filter base   converges to a point   written   if the neigbourhood filter   is contained in the filter   generated by   that is, if   is finer than   In particular, a filter   (which is a filter base that generates itself) converges to   if   Explicitly, to say that a filter base   converges to   means that for every neighbourhood   of   there is a   such that   If a filter base   converges to a point   then   is called a limit (point) of   and   is called a convergent filter base.

A filter base   on   is said to cluster at   (or have   as a cluster point) if and only if each element of   has non-empty intersection with each neighbourhood of   Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an ultrafilter is a limit point.

By definition, every neighbourhood base   at a given point   generates   so   converges to   If   is a filter base on   then   if   is finer than any neighbourhood base at   For the neighborhood filter at that point, the converse holds as well: any basis of a convergent filter refines the neighborhood filter.

See alsoEdit

NotesEdit

  1. ^ Cartan 1937a.
  2. ^ Cartan 1937b.
  3. ^ Igarashi, Ayumi; Zwicker, William S. (16 February 2021). "Fair division of graphs and of tangled cakes". arXiv:2102.08560 [math.CO].
  4. ^ Davey, B. A.; Priestley, H. A. (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. p. 184.
  5. ^ a b Bergman & Hrushovski 1998.
  6. ^ a b c Dugundji 1966, pp. 211–213.
  7. ^ Dolecki & Mynard 2016, pp. 27–29.
  8. ^ Koutras et al. 2021.
  9. ^ Goldblatt, R. Lectures on the Hyperreals: an Introduction to Nonstandard Analysis. p. 32.
  10. ^ Narici & Beckenstein 2011, pp. 2–7.

ReferencesEdit

Further readingEdit