We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new $\alpha$-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts. We provide a definition of $\alpha$-equivalence modulo equational theories that take into account A, C and AC theories. Based on this notion of equivalence, we show that C-unification is finitary and we provide a sound and complete C-unification algorithm, as a first step towards the development of nominal unification modulo AC and other equational theories with permutative properties.

中文翻译：

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关于名义语法和置换不动点

我们基于不动点约束的原始概念，提出了名义项的 alpha 等价关系的新公理化。我们证明了原子和项之间的标准新鲜度关系可以从更原始的置换不动点概念推导出来，并使用这个结果来证明新的 $\alpha$-等价公理化的正确性。这产生了名义统一的新概念，统一问题的解决方案是成对的定点上下文和替换。虽然它可能看起来不像基于新鲜度约束的名义统一子的标准概念那么自然，但是当考虑等式理论时，基于不动点约束的统一子概念表现得更好：例如，名义统一在存在交换性的情况下仍然是有限的，而当使用新鲜度上下文表达统一词时，它变得无限。我们提供了考虑 A、C 和 AC 理论的 $\alpha$-等价模方程理论的定义。基于这个等价的概念，我们证明了 C 统一是有限的，我们提供了一个健全和完整的 C 统一算法，作为发展名义统一模 AC 和其他具有置换性质的方程理论的第一步。