Automorphisms of supermaximal subspaces

Abstract

An infinite-dimensional vector space V ∞ over a recursive field F is called fully effective if V ∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V ∞ has a dependence algorithm , that is a uniformly effective procedure which when applied to x , a 1 ,…, a n , ∈ V ∞ determines whether or not x is an element of { a 1 ,…, a n }* (the subspace generated by { a 1 ,…, a n }). The study of V ∞ , and of its lattice of r.e. subspaces L ( V ∞ ), was introduced in Metakides and Nerode [15]. Since then both V ∞ and L ( V ∞ ) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology. In [15] Metakides and Nerode observed that a study of L ( V ∞ ) may in some ways be modelled upon a study of L ( ω ), the lattice of r.e. sets. For example, they showed how an e -state construction could be modified to produce an r.e. maximal subspace, where M ∈ L ( V ∞ ) is maximal if dim( V ∞ / M ) = ∞ and, for all W ∈ L ( V ∞ ), if W ⊃ M then either dim( W / M ) < ∞ or dim( V ∞ / W ) < ∞. However, some of the most interesting features of L ( V ∞ ) are those which do not have analogues in L ( ω ). Our concern here, which is probably one of the most striking characteristics of L ( V ∞ ), falls into this category. We say M ∈ L ( V ∞ ) is supermaximal if dim( V ∞ / M ) = ∞ and for all W ∈ L ( V ∞ ), if W ⊃ M then dim( W / M ) < ∞ or W = V ∞ . These subspaces were discovered by Kalantari and Retzlaff [13].