The article is dedicated to the properties of 1-types in small ordered theories, specifically focusing on a particular generalization of the concept of a successor function on integers known as a quasi-successor. The article includes examples of 1-types, present in both discrete and dense orders, on which the 2-formula φ(x, y) acts, which is a quasi-successor. For the first time, the concept of quasisuccessor was introduced by the authors in their previous works. The problems of properties and counting the number of countable models of theories with a definable linear order have been studied by many scientists, among them L. Mayer, S.V. Sudoplatov, B.Sh. Kulpeshov, B.S. Baizhanov, A. Alibek, T.S. Zambarnaya, S. Moconja and P. Tanovic. Formulas with properties of succession play an important role in the study of the case of maximal countable models. Using the compactness theorem for a non-isolated 1-type, it is possible to construct a model in which there is an infinite discrete order of the type ω* + ω. On this discrete order, a complete 1-type with the quasi-successor property can be distinguished. A complete description of the subclass of theories having the maximal countable spectrum opens p the possibility of describing a possible countable spectrum for theories with a definable linear order.
Keywords: small theory, discrete order, linear order, 1-types.
