Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Jordan-H\"older with uniqueness for semimodular semilattices   We present a short proof of the Jordan-H\"older theorem with uniqueness for
semimodular semilattice: Given two maximal chains in a semimodular semilattice
of finite height, they both have the same length. Moreover there is a unique
bijection that takes the prime intervals of the first chain to the prime
intervals of the second chain such that the interval and its image are
up-and-down projective.
  The theorem generalizes the classical result that all composition series of a
finite group have the same length and isomorphic factors. Moreover, it shows
that the isomorphism is in some sense unique.

25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn