  The Constraint Satisfaction Problem (CSP) has been intensively studied in
many areas of computer science and mathematics. The approach to the CSP based
on tools from universal algebra turned out to be the most successful one to
study the complexity and algorithms for this problem. Several techniques have
been developed over two decades. One of them is through associating
edge-colored graphs with algebras and studying how the properties of algebras
are related with the structure of the associated graphs. This approach has been
introduced in our previous two papers (A.Bulatov, Local structure of idempotent
algebras I,II. arXiv:2006.09599, arXiv:2006.10239, 2020). In this paper we
further advance it by introducing new structural properties of finite
idempotent algebras omitting type 1 such as separation congruences, collapsing
polynomials, and their implications for the structure of subdirect products of
finite algebras. This paper also provides the algebraic background for our
proof of Feder-Vardi Dichotomy Conjecture (A. Bulatov, A Dichotomy Theorem for
Nonuniform CSPs. FOCS 2017: 319-330).
