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

TL;DR

This paper introduces efficient algorithms for zeta and Möbius transforms in various classes of finite posets and lattices, including semimodular, geometric, and U-labelable posets, reducing computational complexity.

Contribution

It characterizes geometric lattices as those allowing $O(e)$ transforms and extends algorithms to all semimodular and U-labelable posets, broadening applicability.

Findings

Algorithms achieve $O(e)$ complexity in geometric lattices.

Extended algorithms work for all semimodular lattices.

Generalized algorithms apply to U-labelable posets.

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.