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Jordan-H\"older with uniqueness for semimodular semilattices

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arxiv 1908.09912 v2 pith:3SY7KOVN submitted 2019-08-26 math.CO

classification math.CO
keywords semimodularchainfiniteintervalsjordan-hlengthmoreoverolder
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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.

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