Centralizer clones relative to a strong limit cardinal

TL;DR

This paper extends the concept of centralizer clones in universal algebra to a generalized infinitary setting relative to a strong limit cardinal, providing new characterizations and methods for solving centralizer problems.

Contribution

It introduces a generalized framework for centralizer clones relative to a regular cardinal, especially a strong limit cardinal, and offers new characterizations and solution techniques.

Findings

New characterizations of centralizer clones and double centralizer clones.

A novel method for addressing centralizer problems in the infinitary setting.

Positive solutions for centralizer problems in vector spaces and free actions.

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.