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October 31, 2009 / porton

Filter objects

Let U is a set. A filter (on U) \mathcal{F} is by definition a non-empty set of subsets of U such that A,B\in\mathcal{F} \Leftrightarrow A\cap B\in\mathcal{F}. Note that unlike some other authors I do not require \varnothing\notin\mathcal{F}.

For greater clarity I will use filter objects instead of filters. Below I will describe the properties of filter objects without exact definition and the proofs. You can look here for the formalistic behind.

I will denote the set of all filters objects on a set U as \mathfrak{F}. Filter objects are bijectively related with filters by the bijection “\mathrm{up}” from the set of filter objects to the set of filters. A filter object corresponding to principal filter generated by a set A is equal to A. (Thus the set of subsets of U is a subset of \mathfrak{F}.)

Formal definition of filter objects in the framework of ZF is given here. We will not need the exact definition of filter objects, but only the facts that “\mathrm{up}” is a bijection from filter objects to filters and that a filter object corresponding to principal filter generated by a set A is equal to A.

I will define the order on the set of filter objects by the formula \mathcal{A}\subseteq\mathcal{B} \Leftrightarrow \mathrm{up} \mathcal{A} \supseteq \mathrm{up} \mathcal{B} for every filter objects \mathcal{A} and \mathcal{B}. This order well-agrees with the order of sets on U.

\mathfrak{F} with the above defined order is a complete lattice. (See this draft article for a proof.)

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