形式幂级数理想的拓扑闭包及其在拓扑重写理论中的应用
Topological closure of formal power series ideals and application to topological rewriting theory
AI总结：
本文研究形式幂级数理想的拓扑闭包，给出其构造性证明并应用于拓扑重写理论，指出单项式序与次数兼容时无穷合流和拓扑合流等价，且一般情况下前者更强。
摘要：
我们研究形式幂级数理想及其与拓扑重写理论的关系。由于交换形式幂级数代数是扎里斯基环（Zariski rings），其理想对于由不定元生成的极大理想所定义的adic拓扑是闭的。我们提供了这一结果的构造性证明：若给定理想拓扑闭包中的一个形式幂级数，该证明可计算该级数关于理想标准基的余因子表示。我们将此结果应用于拓扑重写理论，其中出现了两个自然的合流概念：拓扑合流和无穷合流。我们给出明确例子说明，一般而言无穷合流是比拓扑合流严格更强的概念。最后，利用理想的拓扑闭包，我们证明在交换形式幂级数的重写理论中，当所考虑的单项式序与次数兼容时，无穷合流与拓扑合流是等价的。
英文摘要：
We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.
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