Sh:971

• Khelif, A., & Shelah, S. (2010). Équivalence élémentaire de puissances cartésiennes d’un même groupe. C. R. Math. Acad. Sci. Paris, 348(23-24), 1241–1244. DOI: 10.1016/j.crma.2010.10.034 MR: 2745331
• Abstract:
  We prove that if $I$ and $J$ are infinite sets and $G$ an abelian torsion group the groups $G^I$ and $G^J$ are elementarily equivalent for the logic $L_{\infty\omega}$. The proof is based on a new and simple property with a Cantor-Bernstein flavour. A criterion applying to non commutative groups allows us to exhibit various groups (free or soluble or nilpotent or ...) $G$ such that for $I$ infinite countable and $J$ uncountable the groups $G^I$ and $G^J$ are not even elementarily equivalent for the $L_{\omega_I \omega}$ logic. Another argument leads to a countable commutative group having the same property.
• published version (4p)

@article{Sh:971,
 author = {Khelif, Anatole and Shelah, Saharon},
 title = {{\'Equivalence \'el\'ementaire de puissances cart\'esiennes d'un m\^eme groupe}},
 journal = {C. R. Math. Acad. Sci. Paris},
 fjournal = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris},
 volume = {348},
 number = {23-24},
 year = {2010},
 pages = {1241--1244},
 issn = {1631-073X},
 mrnumber = {2745331},
 mrclass = {20A15 (03B15 03C75 20K99)},
 doi = {10.1016/j.crma.2010.10.034}
}