Filter (math)

In mathematics, specifically in order theory, reticles and topology, a filter is a special subset of a partially ordained set. A special case frequently used is when the ordered set considered the power set of a set S{displaystyle S}, 2S=P(S)={x:x S!{displaystyle 2^{S}={mathcal {P}}(S)={x:xsubseq S}}}}(i.e., the set made up of all the subsets of S{displaystyle S}), ordered by the inclusion relationship. The dual notion of a filter is ideal.

Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in his book Topologie Générale as an alternative to the similar notion of a network developed in 1922 by E. H. Moore and H. L. Smith.

General definition

A non-empty subset FI was. I was. ∅ ∅ {displaystyle Fneq varnothing } of a partially ordered set (P, ){displaystyle (P,sqsubseteq)} is a filter if the following conditions are given:

1. For each x한 한 F,and한 한 P,x andimplies thatand한 한 F{displaystyle xin F,yin P,xsqsubseteq y{text{ implica que }}yin F} (F{displaystyle F} is a final section).
2. For each x,and한 한 F,There is a certain elementz한 한 F,such asz xandz and{displaystyle x,yin F,{text{ exists certain element}zin F,{text{ such that}zsqsubseq x{text{ and }zsqsubseteq y} (F{displaystyle F} is a filtered set).

A filter is said own if it's not the same as the whole P{displaystyle P} complete.

While the definition above is the most general way to define a filter on "posets" arbitrary, was originally defined only for lattices, in which case the above definition can be characterized by the following equivalent proposition:

A non-empty subset FI was. I was. ∅ ∅ {displaystyle Fneq varnothing } of a reticulate (P, ){displaystyle (P,sqsubseteq)} is a filter, if and only if it is an "upper" set that is closed under finite "meets": {displaystyle sqcap } (minimum), this is, for everything. x,and한 한 F{displaystyle x,yin F}You have to. x and한 한 F{displaystyle xsqcap yin F}.

The smallest filter containing a given element p{displaystyle p} It's a main filter and p{displaystyle p} It's a main element in this situation. The main filter for p{displaystyle p} is given by the whole {x한 한 P:p x!{displaystyle {xin P:psqsubseteq x}} denoted by ↑ ↑ p{displaystyle {uparrow },p}.

The notion of ideal is the dual notion filter, that is, the ideal gets changed all {displaystyle sqsubseteq } for {displaystyle sqsupseteq } and all {displaystyle sqcap } for {displaystyle sqcup } in the filter. Because of this duality the discussion on filters repeats that of ideals. Hence most of the additional information about them (including that of maximal filters and Cousin filters) is found in the article on ideals. There is also a separate article on ultrafilters.

Set Filters

An important case of filters in order theory are the group filters, which are obtained by taking the power set of a given set S{displaystyle S}seen as a partial order and ordered by the inclusion of subsets. With that we will have to have a filter F{displaystyle F} on a set S{displaystyle S} is a set of subsets of S{displaystyle S} with the following properties:

1. SIt's in.F{displaystyle S{text{cH00FF}. (FI was. I was. ∅ ∅ {displaystyle Fneq varnothing } is not empty)
2. F{displaystyle F} does not contain the empty set. (F{displaystyle F} It's his own.)
3. Yeah. AandBThey're in.F,{displaystyle a{text{ and }B{text{ are in }F,} also his intersection. ("F{displaystyle F} is closed under finite intersections ")
4. Yeah. A{displaystyle A} It's in. F{displaystyle F} and A{displaystyle A} is a subset of B{displaystyle B}, then B{displaystyle B} It's in. F{displaystyle F}for all subsets B{displaystyle B} of S{displaystyle S}. (" F{displaystyle F} is closed under supercontents ")

The first three properties imply that a set filter has the Finite Intersection Property. Note that with this definition, a set filter is in effect a filter; in fact it is its own filter. Because of this, it is sometimes called a set's own filter; of course, as clear as it is the context of the array, the shortest name is sufficient.

One filter base It's a subset. B{displaystyle B} of P(S){displaystyle {mathcal {P}}(S)} with the following properties:

1. The intersection of any pair of sets of B{displaystyle B}, contains a set of B{displaystyle B}.
2. B{displaystyle B} is not empty and the empty set, ∅ ∅ {displaystyle varnothing }It's not in. B{displaystyle B}.

Given a filter base B{displaystyle B}, you can get a filter (proper) by including all sets of P(S){displaystyle {mathcal {P}}(S)}containing some subset of B{displaystyle B}. The filter that results is said generated by the filter base B.{displaystyle B.}All filter is to fortiori a filter base, so that the process of moving from a filter base to a filter can be seen as a kind of completeness.

If B and C are two filter bases in S, then C is said to be finer than B (or that C is a refinement of B), if for every B₀ ∈ B exists C₀ ∈ C such that C₀ ⊆ B₀.

For filter bases B and C, if B is finer than C, and C is finer than B, so B and C are said to be equivalent filter bases. Two filter bases are equivalent if and only if the filters they generate are equal.

For filter bases A, B, and C, if A is finer than B, and B is finer than C, A is finer than C. Therefore the refinement relation is a preorder in the set of filter bases, and the transition from a filter base to a filter is an example of a preordering to the associated partial ordering.

Given a subset T of P(S) we can ask when there exists a smaller filter F than contains T. Such a filter exists if and only if the finite intersection of subsets of T is non-empty. We call T subbase of F, and we say that F is generated by T. The subbase T can be constructed by taking all the finite intersections of T, which is then a filter basis for F.

Examples

• Sea (X,T){displaystyle (X,T)} a topological and x{displaystyle x} an element of space X{displaystyle X}, the family of the environments of the point x{displaystyle x} It's a filter over X{displaystyle X}.
• Yeah. A{displaystyle A} is a subset of X{displaystyle X} and x{displaystyle x} an element of the closure A{displaystyle A}, the family {A V:Vis environmentx!{displaystyle {Acap V:V{text{text{}x} It's a filter over A{displaystyle A}.

• Sea S{displaystyle S} an unempty set and C{displaystyle C} a subset of S{displaystyle S} not empty. Then {C!{displaystyle {C}} It's a filter base. The filter it generates (i.e. the collection of all subsets containing to C{displaystyle C}) is called main filter generated by C{displaystyle C}.

• A filter is said to be filter if the intersection of all its elements is empty. A main filter is not free. Since the intersection of any finite number of members of a filter is also a member, no filter on a finite set is free, and in fact is the main filter generated by the common intersection of all its members. A non-main filter on an infinite set is not necessarily free.

• The Fréchet filter of an infinite set S{displaystyle S} is the set of all subsets of S{displaystyle S} that have finite complement. The Frechet filter is free, and is contained in any free filter on S{displaystyle S}.

• A uniform structure on a set X{displaystyle X} is (in particular) a filter in X× × X{displaystyle Xtimes X}.

• A filter in a partially ordered set can be built using the Lema of Rasiowa-Sikorski. almost always used in forcing.