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Algorithmic Content and Structure in Effective Vector Spaces
by
Rumen Dimitrov
Western Illinois University
The two mid-levels of the lattice L(3,Z₂) of subspaces of a 3-dimensional vector space over Z₂ provide a natural model of the Fano plane. We will show that this structure also appears when studying sublattices of infinite dimensional effective vector spaces. In the framework introduced by Metakides and Nerode the study of effective vector spaces can be reduced to the study of a particular computable ℵ₀-dimensional vector space denoted V_{∞}. The vectors in V_{∞} are the finitely nonzero ω-sequences of elements of the underlying computable field F and the operations are pointwise. The space V_{∞} has dependence algorithm and so we can construct a computable basis I₀ of V_{∞}. The lattice of computably enumerable subspaces(c.e.) of V_{∞} is richer than the structure of the lattice of c.e. subsets of I₀. In fact both L(V_{∞}) and L^{∗}(V_{∞}) (which is L(V_{∞}) modulo finite dimension) are non-distributive modular lattices. We will show that if F=Z₂ then for every n there is a particular principal filter in L^{∗}(V_{∞}) which is isomorphic to L(n,Z₂).
Date received: November 13, 2012
Copyright © 2012 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbfw-23.