Semigroups in Stable Structures

Abstract

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