# 2-idempotent 3-quasigroups with a conjugate invariant by Ji L. By Ji L.

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Additional resources for 2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

Example text

Let A be the subgroup generated by the A^, g G G. ,gke G (using minimality of AI, the fact that G' is centreless and finiteness of Morley rank). The proof now follows a series of steps: Let R be the ring of endomorphisms of A generated by G' (acting on A by conjugation) and let I be the ideal in R consisting of those r eR which annihilate AI. (1) R/l is an infinite field K, which is precisely the ring of endomorphisms of AI generated by G'/CG'(AI). 14. (Note CG'(AI) & G' as G' is centreless). Similarly I = {r e R : r(Ax) = 0} is a maximal ideal of R for all g e G.

Before stating and proving this we make a few explanatory remarks: First let G be an co-stable group, A d G and a e G; we say that a is generic over A if tp(a/A) has a nonforking extension peSi(G) which is generic (Note that if p € Si(G) is generic then p does not fork over 0 so V A c: G p [A is "generic"). g. A is a normal subgroup of G and G acts by conjugation). We call X d A Ginvariant if X is fixed setwise by G. 13: Let X c: A be G-invariant. Then X is indecomposable if and only if for every definable G-invariant subgroup H of A we have IX/HI = 1 or oo.

Si on est en caractéristique 0, il est nécessaire que f(â) s'exprime comme R(â), où R(x) est une fraction rationnelle à coefficients dans K; on observe que f(x) = R(x) est une formule A(x) satisfaite par à, c'est-à-dire par le type p. ,An forment une partition de Kn: en caractéristique nulle, à toute application constructible f est associée un découpage de Kn en un nombre fini de parties constructibles sur chacune desquelles f s'exprime rationellement (avec, bien sûr, un dénominateur qui ne s'annulle pas sur l'ensemble en question).