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The equational compatibility problem for the real line. (English) Zbl 1155.03026

Summary: Recently, W. Taylor [Algebra Univers. 55, No. 4, 409–456 (2006; Zbl 1108.03048)] proved that there is no algorithm for deciding of a finite set of equations whether it is topologically compatible with the real line in the sense that it has a model with universe \({\mathbb{R}}\) and with basic operations which are all continuous with respect to the usual topology of the real line. Taylor’s account used operation symbols suitable for the theory of rings with unit together with three unary operation symbols intended to name trigonometric functions supplemented finally by a countably infinite list of constant symbols. We refine Taylor’s work to apply to single equations using operation symbols for the theory of rings with unit supplemented by two unary operation symbols and at most one additional constant symbol.

MSC:

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

Citations:

Zbl 1108.03048
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