Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions
| Autor: | Pentti Haukkanen, Jori Mäntysalo, Mika Mattila |
|---|---|
| Rok vydání: | 2015 |
| Předmět: |
Mathematics::Commutative Algebra
Mathematics::Number Theory Hadamard's maximal determinant problem Semilattice Type (model theory) Theoretical Computer Science law.invention Combinatorics Matrix (mathematics) Invertible matrix Singularity Computational Theory and Mathematics Hadamard transform law Mathematics - Combinatorics Discrete Mathematics and Combinatorics Least common multiple Mathematics |
| Zdroj: | Journal of Combinatorial Theory, Series A. 135:181-200 |
| ISSN: | 0097-3165 |
| Popis: | The invertibility of LCM matrices and their Hadamard powers have been studied a lot over the years by many authors. Bourque and Ligh conjectured in 1992 that the LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots, x_n\}$ is invertible, but in 1997 this was proven false. However, currently there are many open conjectures concerning LCM matrices and their Hadamard powers presented by Hong. In this paper we utilize lattice-theoretic structures and the M\"obius function to explain the singularity of classical LCM matrices and their Hadamard powers. At the same time we end up disproving some of Hong's conjectures. Elementary mathematical analysis is applied to prove that for most semilattice structures there exist a set $S=\{x_1, x_2, \ldots, x_n\}$ of positive integers and a real number $\alpha>0$ such that $S$ possesses this structure and the power LCM matrix $[[x_i, x_j]^\alpha]$ is singular. |
| Databáze: | OpenAIRE |
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