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ω-homomorphisms and ω-groups1

Published online by Cambridge University Press:  12 March 2014

C. H. Applebaum
C. H. Applebaum*
Affiliation:
Bowling Green State University, Bowling Green, Ohio
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Extract

Let ε stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ε (sets), Λ for the set of isols, and for the set of mappings from a subset of ε into ε (functions). I f is a function we write δf and ρf for its domain and range respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β], if δf = α and ρf = β for some function f with a one-to-one partial recursive extension f. We denote the recursive equivalence type of α, {σ ∈ V ∣ ≃ α}, by Req(α). Also R stands for Req(ε), while ΛR denotes the collection of all regressive isols. The reader is assumed to be familiar with the contents of [1] and [6].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 55 - 65
Copyright
Copyright © Association for Symbolic Logic 1971

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Footnotes

1

The results presented in this paper were taken from the author's doctoral dissertation written at Rutgers University under the direction of Professor J. C. E. Dekker.

References

[1]Applebaum, C. H. and Dekker, J. C. E., Partial recursive functions and ω-functions, this Journal (to appear).Google Scholar
[2]Dekker, J. C. E., Good choice sets, Annali della Scuola Normale Superiore di Pisa, Serie III, vol. 20 (1966), pp. 367–393.Google Scholar
[3]Dekker, J. C. E., Infinite series of isols, Proceedings of the Symposium on Recursive Function Theory, American Mathematical Society, Providence, Rhode Island, 1962, pp. 77–96.CrossRefGoogle Scholar
[4]Dekker, J. C. E., and Myhill, J., Recursive equivalence types, University of California Publications of Mathematics (N. S.), vol. 3 (1960), pp. 67–214.Google Scholar
[5]Hall, Marshall, The theory of groups, Macmillan, New York, 1959.Google Scholar
[6]Hassett, M. J., Recursive equivalence types and groups, this Journal, vol. 34 (1969), pp. 13–20.Google Scholar
[7]Nerode, A., Extension to isols, Annals of Mathematics, vol. 73 (1961), pp. 362–403.CrossRefGoogle Scholar
[8]Scott, W. R., Group theory, Prentice Hall, New Jersey, 1964.Google Scholar