Abstract
A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: “equality,” “less,” “between,” “cycle,” and “linkage.” For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description of the automorphism group is given. Although the structure of this lattice was known previously, the proof in the paper is of syntactic nature and avoids the use of a group-theoretical method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. L. Semenov, “Presburgerness of Predicates Regular in Two Number Systems,” Sib. Math. J. 18 (2), 289–300 (1977).
An. A. Muchnik, “The definable criterion for definability in Presburger arithmetic and its applications,” Theoret. Comput. Sci. 290 (3), 1433–1444 (2003).
A. Semenov, S. Soprunov, and V. Uspensky, “The lattice of definability. Origins, recent developments, and further directions,” in Computer Science: Theory and Applications, Lecture Notes in Comput. Sci. (Springer, Cham, 2014), Vol. 8476, pp. 23–38.
P. J. Cameron, “Transitivity of permutation groups on unordered sets,” Math. Z. 148 (2), 127–139 (1976).
C. Frasnay, “Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes,” Ann. Inst. Fourier (Grenoble) 15 (2), 415–524 (1965).
A. L. Semenov and S. F. Soprunov, Lattice of Relational Algebras Definable in Integers with Successor, ar**v: 1201.4439v3 (2012).
A. L. Semenov, “Finiteness conditions for algebras of relations,” Proc. Steklov Inst. Math. 242, 92–96 (2003).
E. V. Huntington, “Inter-relations among the four principal types of order,” Trans. Amer. Math. Soc. 38 (1), 1–9 (1935).
S. A. Adeleke and P. M. Neumann, “Relations related to betweenness: their structure and automorphisms,” in Mem. Amer. Math. Soc. (1998), Vol. 131, No. 623.
P. Erdős and G. Szekeres, “A combinatorial problem in geometry,” Compositio Math. 2, 463–470 (1935).
J. Leroux, “A polynomial time Presburger criterion and synthesis for number decision diagrams,” in 20th Annual IEEE Symposium on Logic in Computer Science (LICS’ 05) (Chicago, IL, 2005), pp. 147–156.
Acknowledgments
The authors are grateful to S. I. Adyan and S. F. Soprunov for discussions and to Yu. A. Boravlev for useful comments and help in the design of the text. We are also grateful to the referees for carefully reading the paper and suggestions concerning its improvement.
Funding
This work was supported in part by the Russian Science Foundation under grant 17-11-01377.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Muchnik, A.A., Semenov, A.L. Lattice of Definability in the Order of Rational Numbers. Math Notes 108, 94–107 (2020). https://doi.org/10.1134/S0001434620070093
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620070093