PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Low
lattice filter (Definition)

Let $ L$ be a lattice. 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. $ F$ is a sublattice of $ L$, and
  2. for any $ a\in F$ and $ b\in L$, $ a\vee b\in F$.

The first condition can be replaced by a weaker one: for any $ a,b\in F$, $ a\wedge b\in F$.

An equivalent characterization of a filter $ I$ in a lattice $ L$ is

  1. for any $ a,b\in F$, $ a\wedge b\in F$, and
  2. for any $ a\in F$, if $ a\le b$, then $ b\in F$.

Note that the dualization switches the meet and join operations, 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 $ F\ne L$, and, if $ L$ contains 0, $ F\ne 0$.
  • $ F$ is a prime filter if it is proper, and $ a\vee b\in F$ implies that either $ a\in F$ or $ b\in F$.
  • $ F$ is an ultrafilter (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\ne\varnothing$ ($ L\in T$), the intersection $ 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 $ \lbrace x\rbrace$, then $ N$ is said to be a principal filter generated by $ x$, written $ [x)$.

Examples.

  1. Consider the positive integers, with meet and join defined by the greatest common divisor and the least common multiple operations. Then the positive even numbers form a filter, generated by $ 2$. If we toss in $ 3$ as an additional element, then $ 1=2\wedge 3\in[\lbrace 2,3\rbrace)$ and consequently any positive integer $ i\in[\lbrace 2,3\rbrace)$, since $ 1\le i$. In general, if $ p,q$ are relatively prime, then $ [\lbrace p,q\rbrace)=\mathbb{Z}^{+}$. In fact, any proper filter in $ \mathbb{Z}^{+}$ is principal. When the generator is prime, the filter is prime, which is also maximal. So prime filters and ultrafilters coincide in $ \mathbb{Z}^{+}$.
  2. Let $ A$ be a set and $ 2^A$ the power set of $ A$. If the set inclusion is the ordering defined on $ 2^A$, 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$.



"lattice filter" is owned by CWoo.
(view preamble)

View style:

See Also: ultrafilter, upper set, lattice ideal, order ideal

Other names:  ultra filter, ultra-filter, maximal filter
Also defines:  filter, prime filter, ultrafilter, filter generated by, principal filter
Log in to rate this entry.
(view current ratings)

Cross-references: ideal, set inclusion, power set, prime, generator, relatively prime, even numbers, least common multiple, greatest common divisor, integers, positive, generated by, singleton, intersection, implies, contains, types, ordering, operations, join, meet, characterization, equivalent, sublattice, subset, lattice
There are 15 references to this entry.

This is version 6 of lattice filter, born on 2006-03-27, modified 2007-07-25.
Object id is 7782, canonical name is LatticeFilter.
Accessed 5578 times total.

Classification:
AMS MSC: 06B10 (Order, lattices, ordered algebraic structures :: Lattices :: Ideals, congruence relations)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy
Least Element by jocaps on 2007-07-23 03:35:16
I am not sure, but I think to define filter of lattices. One does assume that the Lattice has a least element. Although, I do not see why this is necessary.
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)