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arXiv · 1908.09912

Jordan-Hölder with uniqueness for semimodular semilattices

Pavel Paták
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Abstract

We present a short proof of the Jordan-Hölder 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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Pavel Paták. 2019-09-19. Jordan-Hölder with uniqueness for semimodular semilattices. https://arxiv.org/abs/1908.09912

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