Semigroups in stable structures

Yatir Halevi

doi:10.1215/00294527-2018-0003

Semigroups in stable structures

Yatir Halevi

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 1-definable (in M^eq) semigroups S_G;.M/. He also showed that it is strongly -regular: for every p 2 S_G;.M/, there exists n 2 N such that pⁿ is in a subgroup of S_G;.M/. We show that S_G;.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 1-definable semigroups in stable structures.

Original language	English
Pages (from-to)	417-436
Number of pages	20
Journal	Notre Dame Journal of Formal Logic
Volume	59
Issue number	3
DOIs	https://doi.org/10.1215/00294527-2018-0003
State	Published - 1 Jan 2018
Externally published	Yes

Keywords

Epigroup
Newelski’s semigroup
Stable groups
Stable semigroups
Strong pi-regularity

ASJC Scopus subject areas

Logic

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10.1215/00294527-2018-0003

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title = "Semigroups in stable structures",

abstract = "Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG.M/ of complete types concentrated on G is an inverse limit of the 1-definable (in Meq) semigroups SG;.M/. He also showed that it is strongly -regular: for every p 2 SG;.M/, there exists n 2 N such that pn is in a subgroup of SG;.M/. We show that SG;.M/ is in fact an intersection of definable semigroups, so SG.M/ is an inverse limit of definable semigroups, and that the latter property is enjoyed by all 1-definable semigroups in stable structures.",

keywords = "Epigroup, Newelski{\textquoteright}s semigroup, Stable groups, Stable semigroups, Strong pi-regularity",

author = "Yatir Halevi",

note = "Publisher Copyright: {\textcopyright} 2018 by University of Notre Dame.",

year = "2018",

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N2 - Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG.M/ of complete types concentrated on G is an inverse limit of the 1-definable (in Meq) semigroups SG;.M/. He also showed that it is strongly -regular: for every p 2 SG;.M/, there exists n 2 N such that pn is in a subgroup of SG;.M/. We show that SG;.M/ is in fact an intersection of definable semigroups, so SG.M/ is an inverse limit of definable semigroups, and that the latter property is enjoyed by all 1-definable semigroups in stable structures.

AB - Assume that G is a definable group in a stable structure M. Newelski showed that the semigroup SG.M/ of complete types concentrated on G is an inverse limit of the 1-definable (in Meq) semigroups SG;.M/. He also showed that it is strongly -regular: for every p 2 SG;.M/, there exists n 2 N such that pn is in a subgroup of SG;.M/. We show that SG;.M/ is in fact an intersection of definable semigroups, so SG.M/ is an inverse limit of definable semigroups, and that the latter property is enjoyed by all 1-definable semigroups in stable structures.

KW - Epigroup

KW - Newelski’s semigroup

KW - Stable groups

KW - Stable semigroups

KW - Strong pi-regularity

UR - https://www.scopus.com/pages/publications/85055924261

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DO - 10.1215/00294527-2018-0003

M3 - Article

AN - SCOPUS:85055924261

SN - 0029-4527

VL - 59

SP - 417

EP - 436

JO - Notre Dame Journal of Formal Logic

JF - Notre Dame Journal of Formal Logic

IS - 3

ER -

Semigroups in stable structures

Abstract

Keywords

ASJC Scopus subject areas

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