The Bergman-Shelah Preorder on Transformation Semigroups

TL;DR

This paper investigates the structure of a preorder on transformation semigroups of natural numbers, revealing its complexity and its connection to the Continuum Hypothesis, and compares it to similar structures on symmetric groups.

Contribution

It introduces and analyzes a new preorder on transformation semigroups, showing its intricate structure and equivalence to the Continuum Hypothesis, expanding understanding of algebraic and set-theoretic interactions.

Findings

The preorder's structure is highly complex.

A key statement about the preorder is equivalent to the Continuum Hypothesis.

The preorder on semigroups is more complicated than on subgroups of the symmetric group.

Abstract

Let $\nat^\nat$ be the semigroup of all mappings on the natural numbers $\nat$, and let $U$ and $V$ be subsets of $\nat^\nat$. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of $\nat^\nat$ such that $U$ is contained in the subsemigroup generated by $V$ and $C$. We give several results about the structure of the preorder $\preccurlyeq$. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder $\preccurlyeq$ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on $\nat$. The results in this paper suggest that the preorder on subsemigroups of $\nat^\nat$ is much more complicated than that on subgroups of the symmetric group.