Lattice-based and topological representations of binary relations with an application to music.(English)Zbl 1328.00100

Summary: 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.

MSC:

 00A65 Mathematics and music 05E45 Combinatorial aspects of simplicial complexes 06A15 Galois correspondences, closure operators (in relation to ordered sets) 68R05 Combinatorics in computer science

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