ON A QUESTION OF A. RETZLAFF

In [3]: METARIDES and NERODE introduced the study of L(V,), the lattice of recursively enumerable (r.e.) subspaces of an infinite dimensional fully effective vector space V , over a recursive field F . Both KALANTARI [2] and RETZLAFF [6] (amongst others) continued this study. For V , W E L(V,), they defined V to be major in W if V g W , dim(W/V) = co and for R E L(V,) whenever W + R = V , then V + R =* V,, that is, dim(V,/V + 11) < co. One question given by RETZLAFF Let B be a recursive basis of V,. Let M , N E L(B), the lattice of r.e. subsets of B: and suppose M is a major subset of N in L(B). Is M*, the subspzce generat,ed by M , major in N*! In this paper we answer this question affirmatively. We state and prove our results in t-erms of a Steinitz closure system. The study of L ( U ) , the lattice of r.e. closed subsets of an infinite dimensional Steinitz closure system with recursive dependence was introduced in [4], continued in [l] and generalized in [ 5 ] . If the reader is not familiar with 141 or [ 5 ] he is advised to identify U with V , and cl(V) with V*. The obvious advantages of our approach are that the results have a wide range of applications.