A class of ${\Sigma _{3}^{0}}$ modular lattices embeddable as principal filters in ${\mathcal{L}^{\ast }(V_{\infty })}$

Abstract

Let I ₀ be a a computable basis of the fully effective vector space V _∞ over the computable field F. Let I be a quasimaximal subset of I ₀ that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter ${\mathcal{L}^{\ast}(V,\uparrow )}$ of V = cl(I) is isomorphic to the lattice ${\mathcal{L}(n, \overline{F})}$ of subspaces of an n-dimensional space over ${\overline{F}}$ , a ${\Sigma _{3}^{0}}$ extension of F. As a corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices ${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$ is isomorphic to a principal filter of ${\mathcal{ L}^{\ast}(V_{\infty})}$ . We thus answer Question 5.3 “What are the principal filters of ${\mathcal{L}^{\ast}(V_{\infty}) ?}$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets.