  We refine and advance the study of the local structure of idempotent finite
algebras started in [A.Bulatov, The Graph of a Relational Structure and
Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like
structure on an arbitrary finite idempotent algebra omitting type 1. We show
that this graph is connected, its edges can be classified into 3 types
corresponding to the local behavior (semilattice, majority, or affine) of
certain term operations, and that the structure of the algebra can be
`improved' without introducing type 1 by choosing an appropriate reduct of the
original algebra. Then we refine this structure demonstrating that the edges of
the graph of an algebra can be made `thin', that is, there are term operations
that behave very similar to semilattice, majority, or affine operations on
2-element subsets of the algebra. Finally, we prove certain connectivity
properties of the refined structures.
  This research is motivated by the study of the Constraint Satisfaction
Problem, although the problem itself does not really show up in this paper.
