WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
On the Existence of One-Point Time on an Oriented Set
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Abstract: The oriented set notion is the elementary fundamental concept of the theory of changeable sets. In turn, the changeable set theory is closely related to Hilbert's sixth problem. From the formal point of view, any oriented set is a simple relational system with a single reflexive binary relation. Such mathematical structure is the simplest construction, within the framework of which it is possible to give a mathematically strict definition of the time concept. In this regard, the problem of the existence of time with given properties on an oriented set is very interesting. In the present paper, we establish the necessary and sufficient condition for the existence of one-point time on an oriented set. From the intuitive point of view, any one-point time is the time related to the evolution of a system, which consists of a single object (for example, from a single material point). The main result of the paper provides that the one-point time exists on the oriented set if and only if this oriented set is a quasi-chain. Also, using the obtained result, we solve the problem of describing all possible images of linearly ordered sets, which naturally arises in the theory of ordered sets.
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Keywords: Binary relations, reflexive relations, oriented sets, changeable sets, time, ordered sets, quasi-ordered sets
Pages: 1001-1011
DOI: 10.37394/23206.2023.22.109