Profinite Structures are Retracts of Ultraproducts of Finite Structures

Details Profinite Structures are Retracts of Ultraproducts of Finite Structures Profinite Structures are Retracts of Ultraproducts of Finite Structures

2004-01-09

arxiv.org/abs/math/0401095v1

Mathematics

Logic

Scientific paper

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 $\forall \vec{x} (\psi_{0}(\vec{x}) \ra \psi_{1}(\vec{x}))$, 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.

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