可分解的洗牌
Decomposable shuffles
AI总结：
本文提出了一种基于序理论的框架，用于研究可分解的洗牌结构，并通过基本构建块和结构条件在自然数上诱导全序。
摘要：
我们开发了一个组合和序理论的框架，用于洗牌，即作为按索引家庭序列的有序连接，这些序列在自然数上诱导全序。受经典Šarkovskiĭ顺序的启发，我们引入了基本构建块，这些块编码有限和无限顺序模式，并专注于由有限序数以及ω及其对偶ω*构造的可分解洗牌。我们定义了允许在洗牌中定位单个元素的表示，并展示了适当结构条件如何在自然数上产生全序。
英文摘要：
We develop a combinatorial and order-theoretic framework for shuffles, understood as ordered partitions of the natural numbers. We introduce decomposable shuffles as generalised sums of finite ordinals, copies of $ω$, and copies of its dual $ω^*$, indexed by countable scattered linear orders. These building blocks are called benches, ladders and snakes, respectively. We introduce a subclass, layered shuffles, and for this subclass we construct an explicit finite-address representation which reduces comparison in the induced order on $\mathbb N$ to lexicographic comparison of integer tuples. We determine the order types of represented layered shuffles, establish that all represented orders are scattered, and define an involution that realises order duality. Finally, we introduce a hierarchical composition on bench-free injective uniform shuffles of fixed sign. For each sign, this operation yields a monoid graded by reduced degree. Its degree-one component is isomorphic to $\operatorname{Sym}(\mathbb N)$ and consists precisely of the invertible elements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
