Notre Dame Journal of Formal Logic

The Block Relation in Computable Linear Orders

Michael Moses


The block relation B(x,y) in a linear order is satisfied by elements that are finitely far apart; a block is an equivalence class under this relation. We show that every computable linear order with dense condensation-type (i.e., a dense collection of blocks) but no infinite, strongly η-like interval (i.e., with all blocks of size less than some fixed, finite k) has a computable copy with the nonblock relation ¬B(x,y) computably enumerable. This implies that every computable linear order has a computable copy with a computable nontrivial self-embedding and that the long-standing conjecture characterizing those computable linear orders every computable copy of which has a computable nontrivial self-embedding (as precisely those that contain an infinite, strongly η-like interval) holds for all linear orders with dense condensation-type.

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Notre Dame J. Formal Logic, Volume 52, Number 3 (2011), 289-305.

First available in Project Euclid: 28 July 2011

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Zentralblatt MATH identifier

Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
Secondary: 03C57: Effective and recursion-theoretic model theory [See also 03D45]

computable linear order block relation self-embedding


Moses, Michael. The Block Relation in Computable Linear Orders. Notre Dame J. Formal Logic 52 (2011), no. 3, 289--305. doi:10.1215/00294527-1435465.

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  • [1] Ash, C. J., and A. Nerode, "Intrinsically recursive relations", pp. 26–41 in Aspects of Effective Algebra (Clayton, 1979), edited by J. N. Crossley, Upside Down A Book Co., Yarra Glen, 1981.
  • [2] Coles, R. J., R. Downey, and B. Khoussainov, "On initial segments of computable linear orders", Order, vol. 14 (1997/98), pp. 107–24.
  • [3] Downey, R. G., B. Kastermans, and S. Lempp, "On computable self-embeddings of computable linear orderings", The Journal of Symbolic Logic, vol. 74 (2009), pp. 1352–66.
  • [4] Dushnik, B., and E. W. Miller, "Concerning similarity transformations of linearly ordered sets", Bulletin of the American Mathematical Society, vol. 46 (1940), pp. 322–26.
  • [5] Feiner, L., "Hierarchies of Boolean algebras", The Journal of Symbolic Logic, vol. 35 (1970), pp. 365–74.
  • [6] Fellner, S. M., Recursiveness and Finite Axiomatizability of Linear Orderings, ProQuest LLC, Ann Arbor, 1976. Ph.D. thesis, Rutgers The State University of New Jersey, New Brunswick.
  • [7] Gončarov, S. S., and A. T. Nurtazin, "Constructive models of complete decidable theories", Algebra i Logika, vol. 12 (1973), pp. 125–42, 243.
  • [8] Jockusch, C. G., Jr., Reducibilities in Recursive Function Theory, Ph.D. thesis, MIT, Cambridge, 1966.
  • [9] Jockusch, C. G., Jr., and R. I. Soare, "Degrees of orderings not isomorphic to recursive linear orderings, (International Symposium on Mathematical Logic and its Applications, Nagoya, 1988)", Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.
  • [10] Kierstead, H. A., "On $\Pi_1$"-automorphisms of recursive linear orders, The Journal of Symbolic Logic, vol. 52 (1987), pp. 681–88.
  • [11] Moses, M., "Decidable discrete linear orders", The Journal of Symbolic Logic, vol. 53 (1988), pp. 531–39.
  • [12] Moses, M., "Relations intrinsically recursive in linear orders", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 467–72.
  • [13] Rosenstein, J. G., Linear Orderings, vol. 98 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.