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m-normal theories. (English) Zbl 0996.03026

In this paper, the author extends his analysis of multiplicity developed in analogy to geometric stability theory for small stable theories [J. Symb. Log. 57, 644-658 (1992; Zbl 0774.03015); Ann. Pure Appl. Logic 70, 141-175 (1994; Zbl 0817.03017); Fundam. Math. 146, 121-139 (1995; Zbl 0829.03016); and ibid. 150, 149-171 (1996; Zbl 0865.03022); see also Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, Vol. II, 33-42 (1998; Zbl 0907.03015)] to *-algebraic tuples (possibly infinite tuples which are algebraic over a finite set) in arbitrary small countable theories. He defines \(a\) to be m-independent of \(b\) over \(A\) if \(\text{tp}(a/Ab)\) is open in \(\text{tp}(a/A)\); the M-rank is the foundation rank with respect to m-dependence (m-forking), and a theory is m-stable if it has ordinal M-rank. Then m-independence satisfies transitivity, symmetry, non-triviality, and existence; moreover Lascar style inequalities hold for M-rank. A theory is m-normal (which is analogous to one-based) if for any \(a\), \(b\), \(A\) there is a finite \(c\in \text{acl}(Aa)\cap \text{acl}(Ab)\) such that \(a\) and \(b\) are m-independent over \(Ac\). Then in a small m-normal theory M-rank is either finite or \(\infty\) (M-gap conjecture for m-normal theories). Moreover, a small theory \(T\) of finite M-rank is m-normal iff every M-rank 1 *-algebraic type has locally modular geometry and \(T\) has weak (or full) m-coordinatization. These conditions hold in particular if \(T\) is superstable with few countable models.
The author then studies m-normal *-algebraic groups in small theories and shows that they are abelian-by-finite; moreover type-definable subgroups have a subgroup of finite index almost type-definable over \(\emptyset\).
In two subsequent papers [“Small profinite structures”, Trans. Am. Math. Soc. 354, 925-943 (2002; Zbl 0985.03021); “Small profinite groups”, J. Symb. Log. 66, 859-872 (2001; Zbl 0993.03049)] Newelski further develops this analysis in the appropriate context of profinite structures.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
20E18 Limits, profinite groups
03C60 Model-theoretic algebra
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