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What's the meaning 'filtering' and 'chain'? It's about of partially ordered sets. And can you please give me any example?

Definitions:

  1. A preordered set $(I, \leq)$ is directed if every finite subset $F$ of $I$ has an upper bound.

  2. A subset $J$ of a preordered set $(I, \leq)$ is said to be cofinal if for all $i \in I$ there exists $j \in J$ such that $j \geq i$.

  3. A map $f : H \to I$ between two preordered spaces is said to be filtering if for all $i \in I$ there exists $h \in H$ such that $f(k) \geq i$ whenever $k \in H$ satisfies $k \geq h$.

  4. A subset $C$ of $(X, \leq)$ that is totally ordered for the induced preorder is called a chain.

layman
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Moh Jay
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  • The meaning is in the definition. What is unclear to you? – Crostul Feb 25 '15 at 16:57
  • Yes, i do.Especially, I need some examples of each such terms – Moh Jay Feb 25 '15 at 17:04
  • If you want, you can take trivial examples. If you explain what are your doubts, or if you tell us in what context you have seen these definitions, it will be easier to give appropriate examples instead of trivial ones. – Crostul Feb 25 '15 at 17:09
  • OK.The following is a matter that I want to solves.shall you to help.please. – Moh Jay Feb 25 '15 at 17:26
  • Show that a subset C of a preordered space (X, ≤) is a chain if and only if

    • C × C ⊂ A ∪ A−1, where A := {(x, y) : x ≤ y}, A−1 := {(x, y) : (y, x) ∈ A}. 2. Let (I, ≤) be a directed set. Show that if J ⊂ I is not cofinal, then I \ J is cofinal.

    – Moh Jay Feb 25 '15 at 17:26