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Profinite Structures are Retracts of Ultraproducts of Finite Structures

Mariano H. L., Miraglia F.

Reports on Mathematical Logic

2007

No. 42

169-182

We show that if L is a rst-order language with equality, then pronite L-structures, the projective limits of finite L-structures, are retracts of certain ultraproducts of finite L-structures. As a consequence, any elementary class of L-structures axiomatized by L-sentences of the form for all x( 0(~x) -- 1(~x)), where 0(~x); 1(~x) are positive existential L-formulas, is closed under the formation of pronite objects in L-mod, the category of Lstructures and L-homomorphisms. We also mention some interesting applications of our main result to the Theory of Special Groups that have already appeared in the literature.