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Describing hierarchy of concept lattice by using matrix. (English) Zbl 1475.68370

Summary: Concept lattices (also called Galois lattices) are complete ones with the hierarchical order relation of the formal concepts defined by a formal context or Galois connection. In this paper, we present a new of method describing a hierarchy of a finite concept lattice by using a matrix. Given a finite concept lattice \(L\), we introduce Scott topology \(\sigma(L)\) on \(L\) and choose an order of a unique minimal base for \(\sigma(L)\). Then, there is a one-to-one correspondence between the finite topological space \((L, \sigma(L))\) and a proper square matrix with integral entries; thus we obtain a hierarchy-matrix describing the hierarchy of the concept lattice. We explain how to get the information of the hierarchy from the hierarchy-matrix and discuss the relation between the hierarchy-matrix and the Hasse diagram. Since the hierarchy-matrix allowed us to store the information of hierarchy of the concept lattice, we believe that any software autonomously understand the information of hierarchy of the concepts from the hierarchy-matrix.

MSC:

68T30 Knowledge representation
06B23 Complete lattices, completions
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