Necessary and sufficient condition for the existence of the one-point time on an oriented set

• Main Author: Ya.I. Grushka
• Format: Article
• Language: English
• Published: Publishing House "Akademperiodyka" 2019-08-01
• Series: Доповiдi Нацiональної академiї наук України
• Subjects: changeable sets; time; ordered sets; oriented sets
• Online Access: http://www.dopovidi-nanu.org.ua/en/archive/2019/8/2

The subject of this article is closely related to the theory of changeable sets. The mathematically rigorous theory of changeable sets was constructed in 2012. From an intuitive point of view, the changeable sets are sets of objects which, unlike elements of ordinary (static) sets, can be in the process of continuous transformations, i.e., they can change their properties, appear or disappear, and disintegrate into several parts or, conversely, several objects can merge into a single one. In addition, the picture of the evolution of a changeable set can depend on the method of observation (that is, on the reference frame). The main motivation for the introduction of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. The notion of oriented set is the basic elementary concept of the theory of changeable sets. Oriented sets can be interpreted as the most primitive abstract models of sets of changing objects that evolve within a single (fixed) reference frame. The oriented sets are mathematical objects, in the framework of which one can give the mathematically rigorous definition of the concept of time as a certain mapping from a certain time scale, represented by a linearly ordered set, into the set of simultaneous states of the oriented set. In this paper, the necessary and sufficient condition of the existence of the onepoint time on an oriented set is established. From the intuitive point of view, the onepoint time is the time associated with the evolution of a system consisting of only one object (for example, one material point). Namely, the concept of a quasichain oriented set is introduced, and it is proved that the onepoint time exists on the oriented set, if and only if this oriented set is a quasichain. Using the obtained result, the problem of describing all possible images of linearly ordered sets is solved. This problem naturally arises in the theory of ordered sets.