Orbits of maximal vector spaces

R. D. Dimitrov ; V. Harizanov

Algebra i logika, Tome 54 (2015) no. 6, pp. 680-732.

Résumé

Let $V_\infty$ be a standard computable infinite-dimensional vector space over the field of rationals. The lattice $\mathcal L(V_\infty)$ of computably enumerable vector subspaces of $V_\infty$ and its quotient lattice modulo finite dimension subspaces, $\mathcal L^*(V_\infty)$, have been studied extensively. At the same time, many important questions still remain open. R. Downey and J. Remmel [question 5.8, p. 1031, in: Yu. L. Ershov (ed.) et al., Handbook of recursive mathematics. Vol. 2: Recursive algebra, analysis and combinatorics (Stud. Logic Found. Math., 139), Amsterdam, Elsevier, 1998] posed the question of finding meaningful orbits in $\mathcal L^*(V_\infty)$. We believe that this question is important and difficult and its answer depends on significant progress in the structure theory for the lattice $\mathcal L^*(V_\infty)$, and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit of $\mathcal L^*(V_\infty)$. More specifically, we consider two vector spaces, $V_1$ and $V_2$, which are spanned by two quasimaximal subsets of, possibly different, computable bases of $V_\infty$. We give a necessary and sufficient condition for the principal filters determined by $V_1$ and $V_2$ in $\mathcal L^*(V_\infty)$ to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism $\Phi$ of $\mathcal L^*(V_\infty)$ such that $\Phi$ maps the equivalence class of $V_1$ to the equivalence class of $V_2$. Our results are expressed using m-degrees of relevant sets of vectors. This study parallels the study of orbits of quasimaximal sets in the lattice $\mathcal E$ of computably enumerable sets, as well as in its quotient lattice modulo finite sets, $\mathcal E^*$, carried out by R. Soare in [Ann. Math. (2), 100 (1974), 80–120]. However, our conclusions and proof machinery are quite different from Soare's. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space in $\mathcal L^*(V_\infty)$ is generally much more complicated than the one of a principal filter determined by a quasimaximal set in $\mathcal E^*$. We also state that, unlike in $\mathcal E^*$, having isomorphic principal filters in $\mathcal L^*(V_\infty)$ is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of $\mathcal L^*(V_\infty)$.

Détail

Keywords: infinite-dimensional vector space over field of rationals, principal filter, lattice.
Mots-clés : quasimaximal set, equivalence classes, orbit

Bibliographie

[1] A. I. Maltsev, “O rekursivnykh abelevykh gruppakh”, Dokl. AN SSSR, 146:5 (1962), 1009–1012 | Zbl

[2] J. C. E. Dekker, “Countable vector spaces with recursive operations. I”, J. Symb. Log., 34 (1969), 363–387 | DOI | MR | Zbl

[3] G. Metakides, A. Nerode, “Recursively enumerable vector spaces”, Ann. Math. Logic, 11 (1977), 147–171 | DOI | MR | Zbl

[4] R. G. Downey, J. B. Remmel, “Computable algebras and closure systems: coding properties”, Handbook of recursive mathematics, v. 2, Stud. Logic Found. Math., 139, Recursive algebra, analysis and combinatorics, eds. Yu. L. Ershov et al., Elsevier, Amsterdam, 1998, 997–1039 | MR

[5] A. S. Morozov, “Groups of computable automorphisms”, Handbook of recursive mathematics, v. 1, Stud. Log. Found. Math., 138, Recursive model theory, eds. Yu. L. Ershov et al., Elsevier, Amsterdam, 1998, 311–345 | DOI | MR | Zbl

[6] A. Nerode, J. B. Remmel, “A survey of lattices of r.e. substructures”, Recursion Theory, Proc. Symp. Pure Math., 42, eds. A. Nerode, R. Shore, Am. Math. Soc., Providence, RI, 1985, 323–375 | DOI | MR | Zbl

[7] C. J. Conidis, “Infinite dimensional proper subspaces of computable vector spaces”, J. Algebra, 406 (2014), 346–375 | DOI | MR | Zbl

[8] R. G. Downey, D. R. Hirschfeldt, A. M. Kach, S. Lempp, J. R. Mileti, A. Montalbán, “Subspaces of computable vector spaces”, J. Algebra, 314:2 (2007), 888–894 | DOI | MR | Zbl

[9] R. M. Friedberg, “Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication”, J. Symb. Log., 23:3 (1958), 309–316 | DOI | MR

[10] D. A. Martin, “Classes of recursively enumerable sets and degree of unsolvability”, Z. Math. Logik Grundlagen Math., 12:4 (1966), 295–310 | DOI | MR | Zbl

[11] C. F. Kent, “Constructive analogues of the group of permutations of the natural numbers”, Trans. Am. Math. Soc., 104:2 (1962), 347–362 | DOI | MR | Zbl

[12] R. I. Soare, Recursively enumerable sets and degrees, Perspect. Math. Log., Omega Series, Springer-Verlag, Heidelberg a.o., 1987 ; R. I. Soar, Vychislimo perechislimye mnozhestva i stepeni, Kazanskoe matem. ob-vo, Kazan, 2000 | DOI | MR | Zbl | MR

[13] R. I. Soare, “Automorphisms of the lattice of recursively enumerable sets. I: Maximal sets”, Ann. Math. (2), 100 (1974), 80–120 | DOI | MR | Zbl

[14] I. Kalantari, A. Retzlaff, “Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces”, J. Symb. Log., 42 (1977), 481–491 | DOI | MR

[15] J. Remmel, “On r.e. and co-r.e. vector spaces with nonextendible bases”, J. Symb. Log., 45 (1980), 20–34 | DOI | MR | Zbl

[16] D. R. Guichard, “Automorphisms of substructure lattices in recursive algebra”, Ann. Pure Appl. Logic, 25 (1983), 47–58 | DOI | MR | Zbl

[17] G. R. Hird, “Recursive properties of relations on models”, Ann. Pure Appl. Logic, 63:3 (1993), 241–269 | DOI | MR | Zbl

[18] R. G. Downey, G. R. Hird, “Automorphisms of supermaximal subspaces”, J. Symb. Log., 50 (1985), 1–9 | DOI | MR | Zbl

[19] R. D. Dimitrov, “Quasimaximality and principal filters isomorphism between $\mathcal E^*$ and $\mathcal L^*(V_\infty)$”, Arch. Math. Logic, 43:3 (2004), 415–424 | DOI | MR | Zbl

[20] R. D. Dimitrov, “A class of $\Sigma^0_3$ modular lattices embeddable as principal filters in $\mathcal L^*(V_\infty)$”, Arch. Math. Logic, 47:2 (2008), 111–132 | DOI | MR | Zbl

[21] R. Dimitrov, V. Harizanov, R. Miller, K. J. Mourad, “Isomorphisms of nonstandard fields and Ash's conjecture”, Language, life, limits, Proc. 10th conference on computability in Europe CiE 2014 (Budapest, Hungary, June 23–27, 2014), Lect. Notes Comput. Sci., 8493, eds. A. Beckmann et al., Springer, Berlin, 2014, 143–152 | DOI | MR | Zbl

[22] R. D. Dimitrov, “Cohesive powers of computable structures”, Annuaire Univ. Sofia, Fac. Math. Inform., 99 (2009), 193–201 | MR

[23] S. B. Cooper, Computability theory, Chapman Hall/CRC Math. Ser., Chapman and Hall/CRC, Boca Raton, FL, 2004 | MR | Zbl

[24] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga, Novosibirsk, 1999

[25] R. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967 ; Kh. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR | Zbl | MR

[26] R. D. Dimitrov, “Extensions of certain partial automorphisms of $\mathcal L^*(V_\infty)$”, Annuaire Univ. Sofia, Fac. Math. Inform., 99 (2009), 183–191 | MR

[27] E. Artin, Geometric algebra, Intersci. tracts pure appl. math., 3, Intersci. Publ., Inc., New York; Intersci. Publ. Ltd., London, 1957 | MR | Zbl