Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Filter classes of upsets of distributive lattices   Let us say that a class of upward closed sets (upsets) of distributive
lattices is a finitary filter class if it is closed under homomorphic
preimages, intersections, and directed unions. We show that the only finitary
filter classes of upsets of distributive lattices are formed by what we call
$n$-filters. These are related to the finite Boolean lattice with $n$ atoms in
the same way that filters are related to the two-element Boolean lattice:
$n$-filters are precisely the intersections of prime $n$-filters and prime
$n$-filters are precisely the homomorphic preimages of the prime $n$-filter of
non-zero elements of the finite Boolean lattice with $n$ atoms. Moreover,
$n$-filters on Boolean algebras are the only finitary filter classes of upsets
of Boolean algebras generated by prime upsets.

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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas