Mekler's construction preserves CM-triviality

✍ Scribed by Andreas Baudisch

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 468 KB
• Volume: 115
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(01)00088-4

✦ Synopsis

For every structure M of ÿnite signature Mekler (J. Symbolic Logic 46 (1981) 781) has constructed a group G such that for every Ä the maximal number of n-types over an elementary equivalent model of cardinality Ä is the same for M and G. These groups are nilpotent of class 2 and of exponent p, where p is a ÿxed prime greater than 2. We consider stable structures M only and show that M is CM -trivial if and only if G is CM -trivial. Furthermore, we obtain that the free group F2(p; !) in the variety of 2-nilpotent groups of exponent p ¿ 2 with ! free generators has a CM -trivial !-stable theory.