On the Lattices of NP-Subspaces of a Polynomial Time Vector Space over a Finite Field

In this paper, we study the lower semilattice of NP-subspaces of both the standard polynomial time representation and the tally polynomial time representation of a countably infinite dimensional vector space V, over a finite field F. We show that for both the standard and tally representation of V, , there exists polynomial time subspaces U and FV such that U + V is not recursive. We also study the NP analogues of simple and maximal subspaces. We show that the existence of P-simple and NP-maximal subspaces is oracle dependent in both the tally and standard representations of V,. This contrasts with the case of sets, where the existence of NP-simple sets is oracle dependent but NP-maximal sets do not exist. We also extend many results of Nerode and Remmel (1990) concerning the relationship of P bases and NP-subspaces in the tally representation of V, to the standard representation of V,.