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arXiv · 1603.03889

Fast M\"obius inversion in semimodular lattices and U-labelable posets

Petteri Kaski·Jukka Kohonen·Thomas Westerbäck
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Abstract

We consider the problem of fast zeta and M\"obius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and M\"obius transforms can be computed in $O(e)$ elementary arithmetic operations, where $e$ denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in $e$ operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in $e$ operations for all R-labelable posets and their non-graded generalization, which we call U-labelable.

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Petteri Kaski, Jukka Kohonen, Thomas Westerbäck. 2016-03-12. Fast M\"obius inversion in semimodular lattices and U-labelable posets. https://arxiv.org/abs/1603.03889

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