论文标题
相对于强限制的中央式克隆
Centralizer clones relative to a strong limit cardinal
论文作者
论文摘要
通用代数中的运营的换向概念导致了中央器克隆的概念，并引起了我们称之为中央器问题的众所周知的问题，在这些问题中，人们试图确定给定的一组操作是否作为中央器出现，还是等效地与其自身的双重核心器相吻合。几位作者已经研究了通用代数的中央器克隆和中央器问题，Cohn，Kuznecov，Danil'čenko和Harnau的早期贡献。在本文中，我们在相对于常规的基本$α$的无限通用代数的通用环境中工作，从而允许其贫苦却小于$α$的ARITE的操作，并且我们研究了相对于$α$定义的Centralizer Clone的概念。在这种情况下，我们建立了中央器克隆和双中心剂克隆的几个新特征，特别注意$α$是强烈的限制性基本主管，我们讨论了这些结果如何启用一种新的方法来治疗中央器问题。我们将这些结果应用于为几个特定类别的代数结构（包括向量空间，群体的自由行动以及自由单体的自由行动）建立积极的解决方案。
The notion of commutation of operations in universal algebra leads to the concept of centralizer clone and gives rise to a well-known class of problems that we call centralizer problems, in which one seeks to determine whether a given set of operations arises as a centralizer or, equivalently, coincides with its own double centralizer. Centralizer clones and centralizer problems in universal algebra have been studied by several authors, with early contributions by Cohn, Kuznecov, Danil'čenko, and Harnau. In this paper, we work within a generalized setting of infinitary universal algebra relative to a regular cardinal $α$, thus allowing operations whose arities are sets of cardinality less than $α$, and we study a notion of centralizer clone that is defined relative to $α$. In this setting, we establish several new characterizations of centralizer clones and double centralizer clones, with special attention to the case in which $α$ is a strong limit cardinal, and we discuss how these results enable a novel method for treating centralizer problems. We apply these results to establish positive solutions to finitary and infinitary centralizer problems for several specific classes of algebraic structures, including vector spaces, free actions of a group, and free actions of a free monoid.
