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Subspaces of computable vector spaces. (English) Zbl 1127.03036
This paper is continuation of the work begun by a subset of the authors in [R. G. Downey, et al., J. Algebra 314, No. 2, 872–887 (2007; Zbl 1127.03037)]. The authors prove results on the computability theory and reverse mathematics of existence theorems for vector subspaces. They construct a computable vector space $$V$$ of dimension greater than 1 (over a computable field) such that deg$$(W) \gg 0$$ for all nontrivial proper subspaces $$W$$ of $$V$$, and another similar vector space for which deg$$(W) \geq 0^\prime$$ for all finite-dimensional nontrivial proper subspaces $$W$$. They also prove the reverse mathematical analogs of these results, showing that, over RCA$$_0$$, the system WKL$$_0$$ is equivalent to the statement “every vector space of dimension greater than one over an infinite field has a nontrivial proper subspace” and that ACA$$_0$$ is equivalent to “every vector space of dimension greater than one over an infinite field has a finite-dimensional nontrivial proper subspace.” Early work in computable vector spaces includes [J. C. E. Dekker, J. Symb. Log. 34, 363–387 (1969; Zbl 0185.02003)] and early work on reverse mathematics of vector spaces is described in section III.4 of [S. G. Simpson, Subsystems of second order arithmetic. Berlin: Springer (1999; Zbl 0909.03048)].

##### MSC:
 03D45 Theory of numerations, effectively presented structures 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments
##### Keywords:
computable vector space; reverse mathematics; subspace
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##### References:
 [1] Arslanov, Marat M., Some generalizations of a fixed-point theorem, Izv. vyssh. uchebn. zaved. mat., Soviet math. (iz. VUZ), 25, 5, 1-10, (1981), translated in: · Zbl 0523.03029 [2] Dekker, Jacob C.E., Countable vector spaces with recursive operations. I, J. symbolic logic, 34, 363-387, (1969) · Zbl 0185.02003 [3] Downey, Rodney G.; Lempp, Steffen; Mileti, Joseph R., Ideals in computable rings, J. algebra, 314, 872-887, (2007) · Zbl 1127.03037 [4] Metakides, George; Nerode, Anil, Recursively enumerable vector spaces, Ann. math. logic, 11, 147-171, (1977) · Zbl 0389.03019
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