## Notre Dame Journal of Formal Logic

### Semigroups in Stable Structures

Yatir Halevi

#### Abstract

Assume that $G$ 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 $M^{\mathrm{eq}}$) semigroups $S_{G,\Delta}(M)$. He also showed that it is strongly $\pi$-regular: for every $p\inS_{G,\Delta}(M)$, 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 3 (2018), 417-436.

Dates
Accepted: 9 May 2016
First available in Project Euclid: 20 June 2018

https://projecteuclid.org/euclid.ndjfl/1529481617

Digital Object Identifier
doi:10.1215/00294527-2018-0003

Mathematical Reviews number (MathSciNet)
MR3832090

Zentralblatt MATH identifier
06939329

#### Citation

Halevi, Yatir. Semigroups in Stable Structures. Notre Dame J. Formal Logic 59 (2018), no. 3, 417--436. doi:10.1215/00294527-2018-0003. https://projecteuclid.org/euclid.ndjfl/1529481617

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