Perhaps one of the most pleasing phenomena concerning L(co), the lattice of r.e. sets, is SOARE'S [20] remarkable result that any pair M I , M , of maximal r.e. sets automorph, that is. there is an automorphism @ of L ( m ) such that @ ( M I ) = M,.
In recent years. various authors have studied the lattices of r.e. substructures of various r.e. structures. A nice survey of results up to 1983 may be found in NERODE and REJIMEL [IS]. In particular, one area which has received a great deal of attention is the lattice of r.e. subspaces L( V,) of a fully effective vector space V,, and its later generalization t o abstract dependence systems (here see [a, 3,4, 12, 13, 14, IS]). We assume the reader is familiar with these notions, and only give a very brief review of some notation and terminology for V , in 5 1. In [12], METARIDES and NERODE defined the notion of a maximal subspace as T I E L( V,) such that dim( V,/ V ) = 00 and for all Pi' E L(V,) if W r> I', then either dim(W/P) < a3 or dim(V,/W) < a. Naturally, the question arose as to wether or not maximal r.e. subspaces automorph. KALANTARI and RETZLAFF [ I l l provided the answer: they defined V E L ( V,) to be k-thin if (if dim(Ir,,/V) = oc, (ii) there existed Q EL(I',) with dim(V,/&) = k and Q r> V and (iii) for all W E L( i',) if W 2 V then either dim( W / V) < 03 or W 3 &. In particular, (in the case k = 0) they called a 0-thin subspace an r.e. supermaximal subspace and here the definition reads that for all W E L( V,) if W 2 V then either dim( W / V ) < cc or IV = V , .
For each k E cc). KALANTARI and RETZLAFF produced k-thin subspaces and thereby showed that maximal subspaces fall into infinitely many different orbits. Later both KERODE and MILLAR asked whether or not every pair of supermaximal subspaces were in t'he same orbit. GUICHARD [lo] showed that every automorphism of L(V,) was induced by an invertable recursive semilinear transformation of V , , a consequence of which was that a pair of supermaximal subspaces could be in the same orbit only if they had the same (dependence) degrees. Since REMMEL [17] had shown that for any r.e. nonzero degree S, there exist supermaximal
the supermaximal (and k-thin) subspaces fall into infinitely many different orbits. Finally. GUICHARD [lo] then modified a construction of REMMEL 1171 by diagonalizing over recursive invertable semilinear transformations to show that there exist V l , I', E L ( V , ) such that d(V,) = d(D(T',)) = 6 (i = 1, 2 ) and no automorphism of L(B,) takes T', to 7,. This result has been extended in a number of ways by GUICHARD [lo], NERODE and REMMEL [14. 151 and DOWNEY and HIRD [7]. Currently the following (admittedly vague) questions remain open : l ) This research carried out whilst t h e author hold a position at the National University of Singapore. and was partially supported by grant NUS RP-85/83.