幂等代数的局部结构 I
Local structure of idempotent algebras I
AI总结：
本文细化推进了有限幂等代数局部结构研究，引入类图结构并证明其连通性与边的4种类型，针对省略类型1的代数给出可改进结构的约简、细边性质及相关连通性结论，研究受约束满足问题启发。
摘要：
我们对[A.Bulatov，《关系结构的图与约束满足问题》，LICS，2004]中开启的有限幂等代数局部结构研究进行了细化与推进。我们在任意有限幂等代数（包括容许类型1的代数）上引入了一种类图结构。我们证明该图是连通的，其边可被划分为4种类型，分别对应某些项运算的局部行为（集合、半格、多数运算或仿射）。我们还证明，若由该代数生成的簇省略类型1，则可通过选取原代数的合适约简，在不引入类型1的前提下“改进”代数结构；近期提出的Taylor极小幂等代数正是这类约简的一个特例。随后我们细化了这一结构，证明省略类型1的代数的图的边可被设为“细”边，即存在一些项运算，其在代数的二元子集上的行为与半格、多数运算或仿射运算高度相似。最后，我们证明了细化后结构的若干连通性性质。本研究的动机来自约束满足问题（Constraint Satisfaction Problem，CSP）的研究，不过该问题本身并未在本文中真正出现。
英文摘要：
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.
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