# Automorphism groups of substructure lattices of vector spaces

**Date**: Wednesday 29 January 2020, 16:00 – 17:00**Location**: Mathematics Level 8, MALL 1, School of Mathematics**Type**: Logic, Seminars, Pure Mathematics**Cost**: Free

#### 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.

Reference:

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).