Filter Objects

I want to equate principal filters (see below) with the elements of the base poset. Such thing can be done using the principles described in the appendix. The formal definitions follow.

Definition of filter objects

Let \mathfrak{A} is a poset.

Definition Let \uparrow a = \{ x \in \mathfrak{A} | x\supseteq a \} for every a\in\mathfrak{A}. Elements of the set \langle\uparrow\rangle\mathfrak{A} are called principal filters.

Obvious \uparrow is an injection from \mathfrak{A} to \mathfrak{f}.

Let M is a bijection defined on \mathfrak{f} such that M\circ\uparrow = \mathrm{id}_{\mathfrak{A}}. (See the appendix for a proof that such a bijection exists.)

Definition Let \mathfrak{F}=\mathrm{im}M. I call elements of \mathfrak{F} as filter objects (f.o. for short).

Remark Below we will show that \mathrm{up}\mathcal{A} = M^{- 1}\mathcal{A} for each \mathcal{A}\in\mathfrak{F}.

Obvious \uparrow=M^{- 1}|_{\mathfrak{A}}.

Obvious M^{- 1} is a bijection \mathfrak{F}\rightarrow\mathfrak{f}.

Proposition \mathfrak{A}\subseteq\mathfrak{F}.

Proof x\in\mathfrak{A} \Rightarrow M\uparrow x=x \Rightarrow x\in\mathrm{im}M \Rightarrow x\in\mathfrak{F}.

Order of filter objects

Proposition a\subseteq b \Leftrightarrow M^{- 1}a\supseteq M^{- 1}b for all a,b\in\mathfrak{A}.

Proof a\subseteq b \Leftrightarrow \uparrow a\supseteq\uparrow b \Leftrightarrow M^{- 1}a\supseteq M^{- 1}b. QED

As a generalization of the last proposition we may define the order on \mathfrak{F}:

Definition \mathcal{A}\subseteq\mathcal{B} \Leftrightarrow M^{- 1}\mathcal{A}\supseteq M^{- 1}\mathcal{B} for all \mathcal{A},\mathcal{B}\in\mathfrak{A}.

I will call the pair (\mathfrak{F}; \mathfrak{A}) the primary filtrator.

Theorem For the primary filtrator (\mathfrak{F}; \mathfrak{A}) we have \mathrm{up}\mathcal{A}=M^{- 1}\mathcal{A} for each \mathcal{A} \in \mathfrak{F}.

Proof x\in\mathrm{up}\mathcal{A} \Leftrightarrow x\supseteq\mathcal{A} \Leftrightarrow M^{- 1}x\subseteq M^{- 1}\mathcal{A} \Leftrightarrow \uparrow x\subseteq M^{- 1}\mathcal{A} \Leftrightarrow x\in M^{- 1}\mathcal{A} for any x \in \mathfrak{A}. QED

So we have:

  • \mathrm{up} is a bijection from \mathfrak{F} to \mathfrak{f}.
  • \mathcal{A}\subseteq\mathcal{B} \Leftrightarrow \mathrm{up}\mathcal{A}\supseteq\mathrm{up}\mathcal{B} for each \mathcal{A},\mathcal{B}\in\mathfrak{F}.
  • \mathrm{up} a = \uparrow a for every a\in\mathfrak{A}.

A filter object \mathcal{A} is represented by the value of \mathrm{up}\mathcal{A}. We are not interested in the internal structure of filter objects (which can be inferred from the appendix), but only in the value of \mathrm{up}\mathcal{A}. Thus the name “filter objects” by analogy with an object in object oriented programming where an object is completely characterized by its methods, likewise a filter object \mathcal{A} is completely characterized by \mathrm{up}\mathcal{A}.

page_revision: 5, last_edited: 1262018917|%e %b %Y, %H:%M %Z (%O ago)
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