## Abstract.

A (finite or infinite) set ∑ of equations, in operation symbols *F*_{ t } (*t* ∈*T*) and variables *x*_{ i }, is said to be *compatible* with \({\user2{{\mathbb{R}}}}\) iff there exist continuous operations *F* _{ t } ^{ A } on \({\user2{{\mathbb{R}}}}\) such that the algebra \({\mathbf{A}}\, = \,({\user2{{\mathbb{R}}}};\,F^{{\mathbf{A}}}_{t} )_{{t \in T}}\) satisfies the equations ∑ (with the variables *x*_{ i } understood as universally quantified). It is proved that there is no algorithm to decide \({\user2{{\mathbb{R}}}}\)-compatibility for all finite ∑.

If the definition is restricted to *C*^{1} idempotent operations *F* _{ t } ^{ A } , then there *does exist* an algorithm for compatibility.

## 2000 Mathematics Subject Classification.

Primary: 08B05 Secondary: 03D35, 22A30, 26B40, 39B22## Key words and phrases.

topological algebra algorithm compatibility diophantine equation functional equation undecidability differentiability Tarski algorithm [n]-th power variety## Preview

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## Copyright information

© Birkhäuser Verlag, Basel 2007