Results 1 to 5 of 5

Math Help - Filters: Question about Definition

  1. #1
    Newbie
    Joined
    Mar 2012
    From
    Dresden
    Posts
    24
    Thanks
    1

    Filters: Question about Definition

    Hello guys,

    since there is no section "Topology" or "Logic", I think "Advanced Algebra" is yet the closest.
    I've come upon the term "filter" in Mathematics. Principally, I have no problem understanding it - however there is one point in the
    definition that I don't understand. It is as follows:

    Be (P, \leq) a partially ordered set and F a subset. Then F is a filter iff

    1. ...

    2.  \forall x,y \in F \ \exists \ z \in F : z \leq x \ \text{and} \ z \leq y

    3. ...

    Now the question arises with 2.: Why do I need two elements x,y to be greater or equal to z? Why is one not enough, and why do I not need three or something? What exactly does that mean?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: Filters: Question about Definition

    Hey mastermind2007.

    I'm wondering if there are two different kinds of orders here of if the partial order used refers to the same order.

    Also do you know the partial order properties of x with respect to y?

    The z <= x and z <=y implies that x <= y if we are talking about the same partial order. This means that there is a point somewhere where everything is "less than" some particular value of y which means that if a <= y then everything gets filtered on this condition.

    I get the feeling this is needed because of the partial ordering axiom, but I'm not completely sure since this rule is part of the ordering axioms themselves.

    When I read the definition, it seems that all this thing is doing is classifying one subset to be <=, it's complement to be not part of the filter and then keeping the subset that satisfies the partial order with respect to some particular reference point, where the ordering can be whatever the hell it wants to be as long as it divides the set into two groups (filtered and non-filtered) and carries some additional ordering within the filtered subset characterized by the partial order itself.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,501
    Thanks
    765

    Re: Filters: Question about Definition

    Questions about logic should be posted in the Discrete Mathematics forum (its subtitle is "Discrete mathematics, logic, set theory").

    Quote Originally Posted by mastermind2007 View Post
    2.  \forall x,y \in F \ \exists \ z \in F : z \leq x \ \text{and} \ z \leq y

    Now the question arises with 2.: Why do I need two elements x,y to be greater or equal to z? Why is one not enough, and why do I not need three or something?
    The property  \forall x \in F \ \exists z \in F : z \leq x is not a restriction because one can take z = x. On the other hand, it is easy to see by induction using the transitivity of ≤ that if a lower bound of any two elements of F is in F, then a lower bound of any finite number of elements of F is again in F.

    Quote Originally Posted by chiro
    The z <= x and z <=y implies that x <= y if we are talking about the same partial order.
    Why?
    Last edited by emakarov; October 4th 2012 at 02:56 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: Filters: Question about Definition

    The idea was that "if" (I didn't throw in the if statement) x <= y then z <=x and z <=y will refer to to the same subset of the partial order satisfying that constraint.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Filters: Question about Definition

    With partially ordered sets (posets), the archtype example I use is subsets of some set X ordered by inclusion. On that analogy, the filter condition is saying "if two sets are in F, then some subset of their intersection is in F".

    Another prototype example is the integers ordered by divisibility. On that analogy, the filter condition is saying "if two integers are in F, then some divisor of their gcd is in F".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear filters autocorrelation and transfer functions
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 28th 2011, 09:18 AM
  2. Replies: 1
    Last Post: August 26th 2010, 03:18 AM
  3. Hartshorne Definition Question
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: August 5th 2010, 08:39 AM
  4. Question on a definition
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 12th 2009, 09:20 AM
  5. Digital IIR filters
    Posted in the Advanced Applied Math Forum
    Replies: 9
    Last Post: June 30th 2008, 07:04 AM

Search Tags


/mathhelpforum @mathhelpforum