Graphs of finite algebras: maximality, rectangularity, and decomposition

In this paper we continue the study of edge-colored graphs associated with finite idempotent algebras initiated in [Bulatov, “Local structure of idempotent algebras I”, CoRR, abs/2006.09599, 2020.]. We prove stronger connectivity properties of such graphs that will allows us to demonstrate several useful structural features of subdirect products of idempotent algebras such as rectangularity and 2-decomposition.