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α-decompositions of α-spaces

Published online by Cambridge University Press:  12 March 2014

Northrup Fowler III
Northrup Fowler III*
Affiliation:
Hamilton College, Clinton, New York 13323
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Extract

The concept of an α-space was introduced and studied by Dekker in [1] and [2] from which we take our terminology and notation. The reader is assumed to be familiar with their contents. In this paper we are interested in the following conjecture which appears in [2, p. 493]:

(*) S ⊕ C = V and S ∥ C and V an α-space ⇒ both S and C are α-spaces.

It can be simply rephrased as:

(**) The two components of an α-decomposition of an α-space are both α-spaces.

In either form it is relevant to the ideas of α-subspace and α-homomorphism. Dekker [2, T5] has established (*) in the case S (or C) is an isolic α-space. In this paper we will prove (*) in two specific cases, one of which is a generalization of Dekker's result.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 483 - 488
Copyright
Copyright © Association for Symbolic Logic 1976

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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. The author also acknowledges the many helpful conversations with Professor A. B. Manaster.

References

REFERENCES

[1] Dekker, J. C. E., Countable vector spaces with recursive operations. Part I, this Journal, vol. 34 (1969), pp. 363–387.Google Scholar
[2] Dekker, J. C. E., Countable vector spaces with recursive operations. Part II, this Journal, vol. 36 (1971), pp. 477–493.Google Scholar