algebra universalis

, Volume 55, Issue 4, pp 409–456 | Cite as

Equations on real intervals

  • Walter Taylor


A (finite or infinite) set ∑ of equations, in operation symbols F t (tT) 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 C1 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 


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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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