| Résume | In the context of groups of finite Morley rank, Zilber introduced a notion of indecomposable subset and he proved that a family of indecomposable subsets containing the identity generates a definable (connected) group. On this basis, we can relate an algebraic property of the group, to be simple, with a model-theoretic property of its theory, to be aleph1-categorical : a simple group of finite Morlet rank is aleph1-categorical.
In this talk, we will review these notions (indecomposability, simplicity, aleph1-categoricity) in the context of non-associative algebraic structures of finite Morley rank. We will first consider Lie rings before moving to K-loops and "symétrons" in the sense of Poizat; we will see that it is possible to characterize the algebraic simplicity of these structures in terms of the aleph1-categoricity of their theories. |
