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Bases of supermaximal subspaces and Steinitz systems. I

Published online by Cambridge University Press:  12 March 2014

Rod Downey
Rod Downey*
Affiliation:
Monash University, Clayton, Victoria, 3168, Australia Chisholm Institute of Technology, Caulfield, Victoria, 3145, Australia
*
National University of Singapore, Kent Ridge, 0511, Singapore
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Extract

One of the most interesting concepts arising from the study of L(V∞), the lattice of r.e. subspaces of a fully effective vector space of infinite dimension (cf. [6], [7] or [10]), was that of a supermaximal subspace. Supermaximal subspaces of V∞ were those with the fewest possible r.e. superspaces, that is, we say M ∈ L(L∞) is supermaximal if dim(V∞/M) = ∞ and for all Q ∈ L(V∞) with Q ⊇ M, either dim(Q/M) < ∞ or Q = V∞. These spaces were particularly interesting because they had no natural analogue in L(ω), the lattice of r.e. sets. Later Metakides and Nerode [8], Baldwin [1] and the author [2] found that supermaximal substructures occurred in more general settings. In particular, they were found to occur in L(F∞), the lattice of r.e. algebraically closed subfields of F∞ (a recursively presented field of infinite transcendence degree) (cf. [3]). The main tool in these later papers was the concept of a Steinitz (closure) system with recursive dependence (cf. [1], [2], [4] or [8]). We assume familiarity with the definitions and basic results of Metakides and Nerode [8], and only give a brief sketch of some nonstandard facts in §2. If the reader is not familiar with Steinitz systems he is advised to either obtain [1] or [2], or simply identify a Steinitz system (U, cl) with (V∞, *), that is, he should identify U with V∞, and cl(A) with A*, the subspace generated by A.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1146 - 1159
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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