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 including those admitting type 1. We show that this graph is connected, its edges can be classified into 4 types corresponding to the local behavior (set, semilattice, majority, or affine) of certain term operations. We also show that if the variety generated by the algebra omits type 1, then the structure of the algebra can be `improved' without introducing type 1 by choosing an appropriate reduct of the original algebra. Taylor minimal idempotent algebras introduced recently is a special case of such reducts. Then we refine this structure demonstrating that the edges of the graph of an algebra omitting type 1 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.

翻译：我们改进并推进了由[A.Bulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004]开始的关于幂等有限代数局部结构的研究。我们为任意有限幂等代数（包括那些允许类型1的代数）引入了一种类图结构。我们证明了该图是连通的，其边可分为4种类型，分别对应于特定项运算的局部行为（集合型、半格型、多数型或仿射型）。我们还证明，若由该代数生成的簇省略类型1，则可通过选择原代数的适当归约，在不引入类型1的情况下“改进”该代数的结构。最近引入的泰勒极小幂等代数是此类归约的一个特例。随后我们细化了该结构，证明了省略类型1的代数的图的边可以变得“细薄”，即存在某些项运算，它们在代数的2元子集上的行为与半格、多数或仿射运算高度相似。最后，我们证明了这些细化结构的某些连通性性质。本研究受约束满足问题研究的推动，尽管问题本身在本文中并未直接出现。