Formal Concept Analysis (FCA) is considered to be a major formalism for knowledge extraction, reduction, representation and analysis. The core of the mathematical theory of FCA is formal concept lattice.
Determination and description problems on the formal concept lattice are basic ones of FCA. The determination problem is “How can one determine the concept lattice of a given context?” and the description problem is “How can one describe the concept lattice of a given context?”. The most communicative description of the concept lattice is known to be given by Hasse diagrams. However, it is difficult for any information retrieval software to autonomously understand the hierarchy of the concepts from Hasse diagrams. If the hierarchy of the concept lattice is described with a matrix corresponding to Hasse diagrams, any software will be able to autonomously understand the hierarchy of the concepts by the hierarchy-matrix.
In order to obtain such matrix, matrix-correspondence of finite topological spaces obtained by the introduction of Scott topology into the finite concept lattices must be employed and then a new hierarchy-matrix describing the hierarchy of the concept lattice be generated. This idea is based on the matrix-correspondence of finite topological spaces and the fact that Scott topology base is a collection of upper sets of every element. This hierarchy-matrix embodies all the information of the Hasse diagram and is well adapted for use to software.
Pak Chol Hong, a researcher at the Faculty of Applied Mathematics, has proposed a method for generating a new hierarchy-matrix, estimating the connectivity of concepts by the hierarchy-matrix, describing the Hasse diagram via a hierarchy-matrix and generating the hierarchy-matrix via the Hasse diagram in a given finite concept lattice.
The results can also be discussed in universal lattices. If the description of the concept lattices via the hierarchy-matrix is linked up with the determination of the concept lattices, then the formal concept analysis will be used more effectively in several areas of information retrieval, knowledge mining and database management.
Meanwhile, he has proposed that the finite concept lattice is an algebraic lattice and a topological lattice with respect to the Scott topology and the category of finite concept lattices as objects and monotonic mappings as morphisms. These results seem helpful to categorical research into FCA.
More information about this is in his paper “Describing hierarchy of concept lattice by using matrix” presented to SCI Journal “Information Sciences”.
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