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.