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Do filters complementive to a given filter form a complete lattice? ★★

Author(s): Porton

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} $. I will denote $ \mathscr{F} $ the lattice of all filters (on $ U $) ordered by set inclusion.

Let $ \mathcal{A}\in\mathscr{F} $ is some (fixed) filter. Let $ D=\{\mathcal{X}\in\mathscr{F} | \mathcal{X}\supseteq \mathcal{A}\} $. Obviously $ D $ is a bounded lattice.

I will call complementive such filters $ \mathcal{C} $ that:

  1. $ \mathcal{C}\in D $;
  2. $ \mathcal{C} $ is a complemented element of the lattice $ D $.
Conjecture   The set of complementive filters ordered by inclusion is a complete lattice.

Keywords: complete lattice; filter

Posted by porton
updated July 31st, 2009
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Number Theory » Analytic N.T.

Distribution and upper bound of mimic numbers ★★

Author(s): Bhattacharyya

Problem  

Let the notation $ a|b $ denote ''$ a $ divides $ b $''. The mimic function in number theory is defined as follows [1].

Definition   For any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $ divisible by $ \mathcal{D} $, the mimic function, $ f(\mathcal{D} | \mathcal{N}) $, is given by,

$$ f(\mathcal{D} | \mathcal{N}) = \sum_{i=0}^{n}\mathcal{X}_{i}(\mathcal{M}-\mathcal{D})^{i} $$

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

Definition   The number $ m $ is defined to be the mimic number of any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $, with respect to $ \mathcal{D} $, for the minimum value of which $ f^{m}(\mathcal{D} | \mathcal{N}) = \mathcal{D} $.

Given these two definitions and a positive integer $ \mathcal{D} $, find the distribution of mimic numbers of those numbers divisible by $ \mathcal{D} $.

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $ \mathcal{D} $.

Keywords: Divisibility; mimic function; mimic number

Posted by facility_cttb@i...
updated June 20th, 2009
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