Automorphism groups of substructure lattices of vector spaces

Rumen Dimitrov, Western Illinois University. Part of the Logic Seminar Series.

Let V_infty denote a canonical fully effective aleph_0-dimensional vector space over a computable field F.  Let L be the lattice of all subspaces of V_infty, and, for a Turing degree d, let

L_d(V_infty) = {V in L : V is d-computably enumerable}.

By GSL_d, we denote the group of 1-1 and onto semilinear transformations <mu, sigma> such that deg(mu) ≤ d and deg(sigma) ≤ d.  We will prove that:

Theorem 1:  For any pair of Turing degrees a, b
Aut(L_a(V_infty)) embeds into Aut(L_b(V_infty)) if and only if a ≤ b.

Theorem 2:  The degree of the isomorphism type of GSL_d is d''.

This talk is based on joint work with V. Harizanov and A. Morozov.

Dimitrov, R., Harizanov, V., Morozov, A.  "Automorphism groups of substructure lattices of vector spaces in computable algebra."  In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds.) CiE 2016. LNCS, vol. 9709, pp. 251--260. Springer, Heidelberg (2016).