Abstract
The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of the obtained quotient set and their classes. This paper studies the algebraic structure of the induced equivalence classes and characterizes those classes that are convex sublattices of the original concept lattice. Particular attention is given to the reductions removing FCA’s unnecessary attributes. The obtained results will be useful to other complementary reduction techniques, such as the recently introduced procedure based on local congruences.
Highlights
IntroductionInduced by an Attribute Reduction in Formal Concept Analysis (FCA). Mathematics 2021, 9, 565
Two important features of Formal Concept Analysis (FCA), in which the notion of Galois connection is fundamental [13,14,15,16], is that the information contained in a relational dataset can be described in a hierarchic manner by means of a complete lattice [17] and that dependencies between attributes can be determined [18,19,20,21], which is fundamental to applications
With respect to attribute reductions in FCA, we recall the main results related to the induced equivalence relation on the set of concepts of the original concept lattice when we reduce the set of attributes of a formal context
Summary
Induced by an Attribute Reduction in FCA. Mathematics 2021, 9, 565. Two important features of FCA, in which the notion of Galois connection is fundamental [13,14,15,16], is that the information contained in a relational dataset can be described in a hierarchic manner by means of a complete lattice [17] and that dependencies between attributes can be determined [18,19,20,21], which is fundamental to applications In both features, the removal of redundant data has a great impact. We establish a sufficient condition to ensure an equivalence between meet-irreducible concepts in the reduced context and in the original one Under this consideration, we prove that when the original concept lattice is isomorphic to a distributive lattice, the induced equivalence classes by the reduction are always sublattices.
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