  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.
