Lattice decompositions through methods using congruence relations

TL;DR

This paper explores lattice decomposition methods using congruence relations to manage the exponential complexity of concept lattices in data analysis, introducing new decompositions and algorithms.

Contribution

It introduces a new 'reverse doubling construction' for lattice decomposition and provides polynomial algorithms for subdirect decomposition.

Findings

Polynomial algorithms for subdirect decomposition are developed.

A new 'reverse doubling construction' decomposition is proposed.

Theoretical results and proofs for the new decomposition are provided.

Abstract

It is well known by analysts that a concept lattice has an exponential size in the data. Thus, as soon as he works with real data, the size of the concept lattice is a fundamental problem. In this chapter, we propose to investigate factor lattices as a tool to get meaningful parts of the whole lattice. These factor lattices have been widely studied from the early theory of lattices to more recent work in the FCA field. This chapter is divided into three parts. In the first part, we present pieces of lattice theory and formal concept analysis, namely compatible sub-contexts, arrow-closed sub-contexts and congruence relations, all three notions used for the sub-direct decomposition and the doubling convex construction used for the second decomposition, also based on congruence relations. In the second part, the subdirect decomposition into subdirectly irreducible factor is given, polynomial algorithms to compute such a decomposition are given and an example is detailled to illustrate the theory. Then in the third section, a new decomposition named "revese doubling construction" is given. An example is given to explain this decomposition. Theoretical results are given and proofs for the new ones also.