Sh:977

• Shelah, S. (2012). Modules and infinitary logics. In Groups and model theory, Vol. 576, Amer. Math. Soc., Providence, RI, pp. 305–316. arXiv: 1011.3581 DOI: 10.1090/conm/576/11376 MR: 2962893
• Abstract:
  We deal with Abelian groups and R-modules. We consider theories in infinitary logic of the form \mathbb{L}_{\lambda,\theta} of such structures M and prove they have elimination of quantifiers up to positive filtered existential formulas, (so ones defining subgroups of some power of M). However, we demand that we expand by enough individual constants. Hence those theories are stable in the appropriate sense and understood to some extent.

  We thank John Baldwin, who while refereeing a different paper in 2026, brought some problems with this work to this authors attention. Among our revisions, we define a new ideal in §4 and use it to give a shorter proof of the main Lemma in §2 (which has also been slightly changed).

  For the details, see the introduction.

• Version 2026-06-09 (23p) published version (12p)

@incollection{Sh:977,
 author = {Shelah, Saharon},
 title = {{Modules and infinitary logics}},
 booktitle = {{Groups and model theory}},
 series = {Contemp. Math.},
 volume = {576},
 year = {2012},
 pages = {305--316},
 publisher = {Amer. Math. Soc., Providence, RI},
 mrnumber = {2962893},
 mrclass = {03C60 (03C10 03C45 03C75 16B70)},
 doi = {10.1090/conm/576/11376},
 note = {\href{https://arxiv.org/abs/1011.3581}{arXiv: 1011.3581}},
 arxiv_number = {1011.3581}
}