Back to Search
Start Over
Lattice-based and topological representations of binary relations with an application to music
- Source :
- Annals of Mathematics and Artificial Intelligence, Annals of Mathematics and Artificial Intelligence, 2015, 73 (3-4), pp.311-334. ⟨10.1007/s10472-014-9445-3⟩, Annals of Mathematics and Artificial Intelligence, Springer Verlag, 2015, 73 (3-4), pp.311-334. ⟨10.1007/s10472-014-9445-3⟩
- Publication Year :
- 2015
- Publisher :
- HAL CCSD, 2015.
-
Abstract
- International audience; Formal concept analysis associates a lattice of formal concepts to a binary relation. The structure of the relation can then be described in terms of lattice theory. On the other hand Q-analysis associates a simplicial complex to a binary relation and studies its properties using topological methods. This paper investigates which mathematical invariants studied in one approach can be captured in the other. Our main result is that all homotopy invariant properties of the simplicial complex can be recovered from the structure of the concept lattice. This not only clarifies the relationships between two frameworks widely used in symbolic data analysis but also offers an effective new method to establish homotopy equivalence in the context of Q-analysis. As a musical application, we will investigate Olivier Messiaen's modes of limited transposition. We will use our theoretical result to show that the simplicial complex associated to a maximal mode with m transpositions is homotopy equivalent to the (m − 2)–dimensional sphere.
- Subjects :
- [INFO.INFO-PL]Computer Science [cs]/Programming Languages [cs.PL]
Model category
Applied Mathematics
Homotopy
Abstract simplicial complex
Topology
Simplicial homology
homotopy invariance
Simplicial complex
n-connected
formal concept analysis
Q-analysis
Artificial Intelligence
mode of limited transposition
Simplicial set
simplicial complex
Betti numbers
combinatorial classification of harmonies
Mathematics
Simplicial approximation theorem
Subjects
Details
- Language :
- English
- ISSN :
- 10122443 and 15737470
- Database :
- OpenAIRE
- Journal :
- Annals of Mathematics and Artificial Intelligence, Annals of Mathematics and Artificial Intelligence, 2015, 73 (3-4), pp.311-334. ⟨10.1007/s10472-014-9445-3⟩, Annals of Mathematics and Artificial Intelligence, Springer Verlag, 2015, 73 (3-4), pp.311-334. ⟨10.1007/s10472-014-9445-3⟩
- Accession number :
- edsair.doi.dedup.....ecd5528f9982c492684a7971f2b946a0