### Abstract

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Mobius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Mobius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) monotone circuits for several lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.

Original language | English |
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Article number | 4 |

Number of pages | 19 |

Journal | ACM Transactions on Algorithms |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2016 |

MoE publication type | A1 Journal article-refereed |

Event | ACM-SIAM Symposium on Discrete Algorithms - Kyoto, Japan Duration: 17 Jan 2012 → 19 Jan 2012 Conference number: 23 |

### Keywords

- Arithmetic circuit
- fast multiplication
- lattice
- zeta transform
- Mobius transform
- Mobius inversion
- semigroup algebra
- SEMIGROUP REPRESENTATION-THEORY
- FAST FOURIER-TRANSFORMS
- HYPERPLANE ARRANGEMENTS
- MOBIUS FUNCTIONS
- ALGEBRAS
- GRAPHS
- SET

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## Cite this

*ACM Transactions on Algorithms*,

*12*(1), [4]. https://doi.org/10.1145/2629429