lattice filter

Let L be a latticeMathworldPlanetmathPlanetmath. A filter (of L) is the dual concept of an ideal ( Specifically, a filter F of L is a non-empty subset of L such that

  1. 1.

    F is a sublattice of L, and

  2. 2.

    for any aF and bL, abF.

The first condition can be replaced by a weaker one: for any a,bF, abF.

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath characterizationMathworldPlanetmath of a filter I in a lattice L is

  1. 1.

    for any a,bF, abF, and

  2. 2.

    for any aF, if ab, then bF.

Note that the dualization switches the meet and join operationsMathworldPlanetmath, as well as reversing the ordering relationship.

Special Filters. Let F be a filter of a lattice L. Some of the common types of filters are defined below.

  • F is a proper filter if FL, and, if L contains 0, F0.

  • F is a prime filter if it is proper, and abF implies that either aF or bF.

  • F is an ultrafilterMathworldPlanetmath (or maximal filter) of L if F is proper and the only filter properly contains F is L.

  • filter generated by a set. Let X be a subset of a lattice L. Let T be the set of all filters of L containing X. Since T (LT), the intersectionMathworldPlanetmath N of all elements in T, is also a filter of L that contains X. N is called the filter generated by X, written [X). If X is a singleton {x}, then N is said to be a principal filterPlanetmathPlanetmath generated by x, written [x).


  1. 1.

    Consider the positive integers, with meet and join defined by the greatest common divisorMathworldPlanetmath and the least common multipleMathworldPlanetmath operations. Then the positive even numbersMathworldPlanetmath form a filter, generated by 2. If we toss in 3 as an additional element, then 1=23[{2,3}) and consequently any positive integer i[{2,3}), since 1i. In general, if p,q are relatively prime, then [{p,q})=+. In fact, any proper filter in + is principal. When the generator is prime, the filter is prime, which is also maximal. So prime filters and ultrafilters coincide in +.

  2. 2.

    Let A be a set and 2A the power setMathworldPlanetmath of A. If the set inclusion is the ordering defined on 2A, then the definition of a filter here coincides with the ususal definition of a filter ( on a set in general.

Remark. If F is both a filter and an ideal of a lattice L, then F=L.

Title lattice filter
Canonical name LatticeFilter
Date of creation 2013-03-22 15:49:01
Last modified on 2013-03-22 15:49:01
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 06B10
Synonym ultra filter
Synonym ultra-filter
Synonym maximal filter
Related topic Ultrafilter
Related topic UpperSet
Related topic LatticeIdeal
Related topic OrderIdeal
Defines filter
Defines prime filter
Defines ultrafilter
Defines filter generated by
Defines principal filter