## Reports on Mathematical Logic

### 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.