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Reports on Mathematical Logic

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No. 42

Hugo Luiz MARIANO and Francisco MIRAGLIA

Profinite Structures are Retracts of Ultraproducts of Finite Structures
A b s t r a c t. 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 $\all \vec{x} (\psi_{0}(\vec{x})
\ra\psi_{1}(\vec{x}))$, where $\psi_{0}(\vec{x}),\psi_{1}(\vec{x})$ 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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