Bulletin of Symbolic Logic

Computable and continuous partial homomorphisms on metric partial algebras

Viggo Stoltenberg-Hansen and John V. Tucker

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Abstract

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations.

Article information

Source
Bull. Symbolic Logic, Volume 09, Issue 3 (2003), 299- 334.

Dates
First available in Project Euclid: 17 July 2003

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1058448675

Digital Object Identifier
doi:10.2178/bsl/1058448675

Mathematical Reviews number (MathSciNet)
MR2005952

Zentralblatt MATH identifier
1058.03070

Citation

Stoltenberg-Hansen, Viggo; Tucker, John V. Computable and continuous partial homomorphisms on metric partial algebras. Bull. Symbolic Logic 09 (2003), no. 3, 299-- 334. doi:10.2178/bsl/1058448675. https://projecteuclid.org/euclid.bsl/1058448675


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