¹Kazakh British Technical University, Almaty, Kazakhstan

²Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

³Novosibirsk State Technical University, Novosibirsk, Russian Federation

⁴Sobolev Institute of Mathematics, Novosibirsk, Russian Federation

One of the important characteristics of structures is degrees of semantic and syntactic rigidity, as well as indices of rigidity, showing how much the given structure differs from semantically rigid structures, i.e., structures with one-element automorphism groups, as well as syntactically rigid structures, i.e., structures covered by definable closure of the empty set. Issues of describing the degrees and indices of rigidity represents interest both in a general context and in relation to ordering theories and their models. In the given paper, we study possibilities for semantic and syntactic rigidity for ordered theories, i.e., the rigidity with respect to automorphism group and with respect to definable closure. We describe values for indices and degrees of semantic and syntactic rigidity for well-ordered sets, for discrete, dense, and mixed orders and for countable models of ℵ₀-categorical weakly o-minimal theories. All possibilities for degrees of rigidity for countable linear orderings are described.

Beibut Sh. Kulpeshov, Dr. Sci. (Phys.-Math.), Prof., Kazakh British Technical University, Almaty, 050000, Kazakhstan, b.kulpeshov@kbtu.kz; Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, kulpesh@mail.ru; Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, kulpeshov@corp.nstu.ru

Sergey V. Sudoplatov, Dr. Sci. (Phys.–Math.), Prof., Sobolev Institute of Mathematics, Novosibirsk, 630090, Russian Federation, sudoplat@math.nsc.ru; Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, sudoplatov@corp.nstu.ru

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