Profinite Structures are Retracts of Ultraproducts of Finite Structures

We show that if L is a first-order language with equality, thenprofinite 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 $\all \vec{x} \ra\psi_{1})$, where $\psi_{0},\psi_{1}$ are positive existential L-formulas, is closed under the formation of profinite objects in L-mod, the category of L-structures 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
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