On the Size of $\exists$-Generalized Concepts

TL;DR

This paper investigates how the size of concept lattices in Formal Concept Analysis can increase after generalizing attributes, providing examples where the lattice size grows more than expected after combining just two attributes.

Contribution

It introduces a family of contexts demonstrating that $orall$-generalization on two attributes can significantly increase the concept lattice size, challenging previous assumptions.

Findings

Size of concept lattice can increase after attribute generalization

Counterexamples show more than one new concept can be created

Generalization effects are context-dependent and can be substantial

Abstract

Formal Concept Analysis (FCA) offers several tools for qualitative data analysis. One possibility is to group objects that share common attributes together and get a concept lattice that describes the data. Quite often the size of this concept lattice is very large. Many authors have investigated methods to reduce the size of this lattice. In \cite{KMBV14} the authors consider putting together some attributes to reduce the size of the attribute sets. But this reduction does not always carry over the set of concepts. They have provided some counter examples where the size of the concept lattice increases by one after putting two attributes together. Then they asked the following question: "How many new concepts can be generated by an $\exists$-generalization on just two attributes?" The present paper provides a family of contexts for which the size increases on more than one concept after putting solely two attributes together.