Taylor, Walter Equations on real intervals. (English) Zbl 1108.03048 Algebra Univers. 55, No. 4, 409-456 (2006). 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. Cited in 2 ReviewsCited in 3 Documents 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 Keywords:topological algebra; algorithm; compatibility; Diophantine equation; functional equation; undecidability; differentiability; Tarski algorithm; [n]-th power variety PDFBibTeX XMLCite \textit{W. Taylor}, Algebra Univers. 55, No. 4, 409--456 (2006; Zbl 1108.03048) Full Text: DOI