Profinite Structures are Retracts of Ultraproducts of Finite Structures

Abstract

We establish the following model-theoretic characterization: profinite $L$-structures, the cofiltered limits of finite $L$-structures,are retracts of ultraproducts of finite $L$-structures. As a consequence, any elementary class of $L$-structures axiomatized by $L$-sentences of the form $\forall \vec{x} (ψ_{0}(\vec{x}) \ra ψ_{1}(\vec{x}))$, where ${\psi }_0(\vec{x}),{\psi }_1(\vec{x})$ are existencial-positives $L$-formulas, is closed under the formation of profinite objects in the category {\bf L-mod}, the category of structures suitable for the language $L$ and $L$-homomorphisms.