Describing hierarchy of concept lattice by using matrix

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 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. (C) 2020 Published by Elsevier Inc.

Automated summary

The paper presents a new method for describing the hierarchy of a finite concept lattice using a matrix, establishing a one-to-one correspondence between a finite topological space and a proper square matrix with integral entries. The hierarchy-matrix allows for storage and understanding of hierarchy information of the concept lattice, facilitating autonomous interpretation by software.