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

1. ...

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

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?

2. ## 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.

3. ## Re: Filters: Question about Definition

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

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

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 $\displaystyle \forall x \in F \ \exists z \in F : z \leq x$ 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.

Originally Posted by chiro
The z <= x and z <=y implies that x <= y if we are talking about the same partial order.
Why?

4. ## 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.

5. ## 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".