Let I be a quasimaximal subset of a computable basis of the fully efective vector space V ∞ . We give a necessary and sufficient condition for the existence of an isomorphism between the principal filter respectivelly. We construct both quasimaximal sets that satisfy and quasimaximal sets that do not satisfy this condition. With the latter we obtain a negative answer to Question 5.4 posed by Downey and Remmel in [3].

Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual presentation of$\omega $, then every cohesive power of$\mathcal {L}$has order-type$\omega + (...) \zeta \eta $. However, there are computable copies of$\omega $, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$\omega + \zeta \eta $. For example, we show that there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \eta $. Our most general result is that if$X \subseteq \mathbb {N} \setminus \{0\}$is a Boolean combination of$\Sigma _2$sets, thought of as a set of finite order-types, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$, where$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$denotes the shuffle of the order-types inXand the order-type$\omega + \zeta \eta + \omega ^*$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X)$. (shrink)

We construct a computable vector space with the trivial computable automorphism group, but with the dependence relations as complicated as possible, measured by their Turing degrees. As a corollary, we answer a question asked by A.S. Morozov in [Rigid constructive modules, Algebra and Logic, 28 570–583 ; 379–387 ].

Let I0 be a a computable basis of the fully effective vector space V∞ over the computable field F. Let I be a quasimaximal subset of I0 that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}^{\ast}(V,\uparrow )}$$\end{document} of V = cl(I) is isomorphic to the lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}(n, \overline{F})}$$\end{document} of subspaces (...) of an n-dimensional space over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{F}}$$\end{document}, a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma _{3}^{0}}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$$\end{document} is isomorphic to a principal filter of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ L}^{\ast}(V_{\infty})}$$\end{document}. We thus answer Question 5.3 “What are the principal filters of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}^{\ast}(V_{\infty})?}$$\end{document} ” 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. (shrink)