Centralizer clones relative to a strong limit cardinal

Abstract

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'\v{c}enko, and Harnau. In this paper, we work within a generalized setting of infinitary universal algebra relative to a regular cardinal $\alpha$, thus allowing operations whose arities are sets of cardinality less than $\alpha$, and we study a notion of centralizer clone that is defined relative to $\alpha$. In this setting, we establish several new characterizations of centralizer clones and double centralizer clones, with special attention to the case in which $\alpha$ 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.