## Abstract

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius 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 Möbius 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 - 1 Dec 2015 |

## Keywords

- Arithmetic circuit
- Fast multiplication
- Lattice
- Möbius inversion
- Möbius transform
- Semigroup algebra
- Zeta transform