ISSN 0001-4346, Mathematical Notes, 2020, Vol. 108, No. 1, pp. 94–107. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 108, No. 1, pp. 102–118. Lattice of Definability in the Order of Rational Numbers An. A. Muchnik †1 and A. L. Semenov 2, 3, 4, 5* 1 Computing Center of Russian Academy of Sciences, Moscow, 119333 Russia 2 Lomonosov Moscow State University, Moscow, 119899 Russia 3 Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333 Russia 4 Axel Berg Institute of Cybernetics and Educational Computing FRC CSC of Russian Academy of Sciences, Moscow, 119333 Russia 5 Moscow Institute of Physics and Technology (National Research University), Moscow Oblast, Dolgoprudny, 141701 Russia Received February 20, 2019; in final form, July 22, 2019; accepted October 29, 2019 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. DOI: 10.1134/S0001434620070093 Keywords: finitely generated space, lattice of definability subspaces, syntactic nature. INTRODUCTION History of the Problem In the early 1970s, Albert Abramovich Muchnik (a disciple of P. S. Novikov) attracted the attention of his disciple A. L. Semenov (one of the authors of the present paper) to the circle of problems related to definability in fragments of arithmetic. An example of the result of this period is a theorem on the definability in the arithmetic of addition of integers of any relation definable in an automaton way in two number systems with multiplicatively independent bases [1], [2]. The problem of finding a complete description of all kinds of weakenings of the arithmetic of addition of integers, or, in other words, of definability subspaces, or reducts (see [3]) was posed (in the sense of expressive power; for the formal definition, see below). In their student works in the 1970s–1980s, L. Kostyukov and O. Mitina advanced in solving the above problem and supposedly found such a complete description. However, their proofs were incomplete. Trying to finalize the solution, the authors of the present paper considered the substantially simpler case of the order on rational numbers instead of the addition of integers in the early 2000s. A technique was developed, which is also useful, as we hope, in other situations (the technique is presented in this publication), and a complete description of “weakenings” for the order of rational numbers was obtained. Such a description can also be obtained in another way. In particular, it follows from constructions of Cameron, in which group-theoretic and model-theoretic methods were used (see [4]), although a description is not explicitly formulated in the paper. Apparently, the result itself was obtained for the first time in [5] and remained unknown to Cameron. To obtain the result, one can also use the Svenonius theorem [6]. For partial advances in solving Muchnik’s initial problem, see also [6] (the case of succession on integers). † Deceased. * E-mail: alsemno@ya.ru 94