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An Ergodic Theory of Binary Operations—Part I: Key Properties

机译：遍历二元运算理论—第一部分：关键属性

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摘要

An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arıkan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (first) part of this paper introduces the mathematical framework that we will use in the second part to characterize the polarizing operations. We define uniformity preserving, irreducible, ergodic, and strongly ergodic operations, and we study their properties. The concepts of a stable partition and the residue of a stable partition are introduced. We show that an ergodic operation is strongly ergodic if and only if all its stable partitions are their own residues. We also study the products of binary operations and the structure of their stable partitions. We show that the product of a sequence of binary operations is strongly ergodic if and only if all the operations in the sequence are strongly ergodic. In the second part of this paper, we provide a foundation of polarization theory based on the ergodic theory of binary operations that we develop in this part.
机译：极化理论中的一个开放问题是确定在Arıkan样式构造中使用时总是导致极化的二进制运算（在一般的多级意义上）。本文分为两部分，通过为二元运算极化提供必要和充分的条件，从而解决了该问题。本文的第一部分介绍了数学框架，我们将在第二部分中使用该框架来描述偏振操作。我们定义了保持均匀性，不可还原，遍历和强遍历的操作，并研究了它们的性质。介绍了稳定分区的概念和稳定分区的残差。我们证明，当且仅当其所有稳定分区都是其自己的残差时，遍历操作才是高度遍历的。我们还研究了二进制运算的乘积及其稳定分区的结构。我们证明，当且仅当该序列中的所有操作都是高度遍历时，二元运算序列的乘积才是高度遍历遍历的。在本文的第二部分中，我们基于在本部分中开发的二元运算的遍历遍历理论提供了极化理论的基础。

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