Semigroups in Stable Structures

TL;DR

This paper investigates the structure of semigroups associated with definable groups in stable structures, showing they are intersections of definable semigroups and extending properties known for $igwedge$-definable semigroups.

Contribution

It proves that $S_{G, riangle}(M)$ is an intersection of definable semigroups, generalizing the inverse limit structure to all $igwedge$-definable semigroups in stable structures.

Findings

$S_{G, riangle}(M)$ is an intersection of definable semigroups

The inverse limit property extends to all $igwedge$-definable semigroups in stable structures

Semigroups in stable structures exhibit strong regularity properties

Abstract

Assume $G$ is a definable group in a stable structure $M$. Newelski showed that the semigroup $S_G(M)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$-definable (in $M^{eq}$) semigroups $S_{G,\Delta}(M)$. He also shows that it is strongly $\pi$-regular: for every $p\in S_{G,\Delta}(M)$ there exists $n\in\mathbb{N}$ such that $p^n$ is in a subgroup of $S_{G,\Delta}(M)$. We show that $S_{G,\Delta}(M)$ is in fact an intersection of definable semigroups, so $S_G(M)$ is an inverse limit of definable semigroups and that the latter property is enjoyed by all $\infty$-definable semigroups in stable structures.