Simplicity theory (Q2846559)

Simplicity theory is the extension of stability theory, i.e.\ the study of Shelah's notion of independence and forking in stable structures, to the wider class of simple theories, which can be defined as those theories where forking independence is symmetric. Its applicational interest stems from the fact that certain natural unstable structures, such as the random graph, vector spaces over finite fields with a non-degenerate bilinear form, pseudofinite fields or fields with a generic automorphism, although not stable are at least simple. In fact, the detailed study of these structures and their stable-like behaviour has very much influenced the development of simplicity theory. The principal aim of the subject is to recover as much as possible from stability theory, in a technically much more involved setting where types have unboundedly many non-forking extensions, canonical bases are no longer imaginary but hyperimaginary (classes of infinite tuples modulo type-definable equivalence relations), and no local theory (formula-by-formula) exists. From a pure point of view, simplicity theory allows to separate phenomena due to the symmetry of independence from phenomena due to bounded multiplicity, or due to the lack of the independence property (as stable theories are precisely the simple theories without the independence property).NEWLINENEWLINEThis is the third monograph on simplicity theory, after the reviewer's [Simple theories. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0948.03032)] and \textit{E. Casanovas}' [Simple theories and hyperimaginaries. Cambridge: Cambridge University Press; Ithaca, NY: Association of Symbolic Logic (ASL) (2011; Zbl 1243.03045)]. Unlike the latter book, which focusses in detail on the pure part of the theory and culminates with the proof of elimination of hyperimaginaries in simple theories, the author of the book under review aims to present practically all the important results of the subject in the first six chapters, and concludes with two chapters on some more recent geometric simplicity theory due in good part to the author himself. Among the few major topics not covered are the simple indecomposability theorem, the binding group, and supersimplicity of unidimensional simple theories.NEWLINENEWLINEChapter 1 summarizes basic model and stability theory; Chapter 2 introduces dividing and forking, proves the equivalence between simplicity, local character, symmetry, transitivity, and dividing with respect to all Morley sequences. Local ranks, Shelah D-rank and Lascar SU-rank are introduced, supersimplicity and lowness are defined, various examples of simple structures are given, and it is shown that a supersimple theory with a finite number of non-isomorphic countable models is \(\aleph_0\)-categorical. In Chapter 3 Lascar strong types are introduced, the amalgamation theorem is shown, and simplicity is characterized by the independence properties together with amalgamation. All of this is generalized to hyperimaginaries in Chapter 4. In Chapter 5, the Lascar group is introduced, and elimination of finitary hyperimaginaries in small theories, as well as full elimination in supersimple theories is shown. Chapter 6 defines one-basedness and CM-triviality, and presents Hrushovski's construction of an \(\aleph_0\)-categorical supersimple non-one-based structure, as well as Chatzidakis' and Pillay's theory of a stable structure with a generic predicate or automorphism, and Vassiliev's \textit{lovely pairs}. Chapter 7 deals with groups: genericity is developed first for type-definable and then hyperdefinable groups, stabilizers and locally connected components are defined, and one-based groups are characterized. It is shown that supersimple type-definable groups are an intersection of definable groups, and that an \(\aleph_0\)-categorical supersimple group is finite-by-abelian-by-finite of finite SU-rank. Finally, a group and a group action is reconstructed from a generically given one.NEWLINENEWLINEThe last two chapters deal with geometric simplicity theory. In Chapter 8 the geometry of an SU-rank 1 type is studied; it is shown that a modular geometry is either trivial or a projective geometry over a division ring. An example for a locally modular non-affine geometry is given, and it is shown how to complete a locally modular geometry \(D\) to a modular geometry in \(D^{\mathrm{heq}}\). Finally, in the last chapter a hyperimaginary group configuration theorem is shown under a generalized amalgamation hypothesis, noting that without this assumption one merely obtains an almost hyperdefinable group.NEWLINENEWLINEThe book covers an enormous amount of material on only 224 pages (of small print). Consequently, many of the proofs are very concise and require an additional effort from the reader. It is written in classical definition-lemma-theorem style, and relatively few explanation (as opposed to proof) is given of what actually is happening. Nevertheless, the approach is streamlined, quick and complete, which makes it an excellent companion for anyone working in simplicity theory.