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Equations on real intervals. (English) Zbl 1108.03048

Summary: A (finite or infinite) set \(\Sigma\) of equations, in operation symbols \(F_t\) (\(t \in T\)) and variables \(x_i\), is said to be compatible with \(\mathbb R\) iff there exist continuous operations \(F_t^{\mathbf A}\) on \(\mathbb R\) such that the algebra \(\mathbf A = (\mathbb R;F_t^{\mathbf A})_{t\in T}\) satisfies the equations \(\Sigma\) (with the variables \(x_i\) understood as universally quantified). It is proved that there is no algorithm to decide \(\mathbb R\)-compatibility for all finite \(\Sigma\).
If the definition is restricted to \(C^1\) idempotent operations \(F_t^{\mathbf A}\), then there does exist an algorithm for compatibility.

MSC:

03D35 Undecidability and degrees of sets of sentences
08B05 Equational logic, Mal’tsev conditions
22A30 Other topological algebraic systems and their representations
26B40 Representation and superposition of functions
39B22 Functional equations for real functions
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