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

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. Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Ya.I. Grushka [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch ; Russisch ; Ukrainisch

Erschienen:	2019

Schlagwörter:	changeable sets time ordered sets oriented sets

Übergeordnetes Werk:	In: Доповiдi Нацiональної академiї наук України - Publishing House "Akademperiodyka", 2019, 8(2019), Seite 9-15
Übergeordnetes Werk:	volume:8 ; year:2019 ; pages:9-15

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc Journal toc

DOI / URN:	10.15407/dopovidi2019.08.009

Katalog-ID:	DOAJ008124957

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spelling	10.15407/dopovidi2019.08.009 doi (DE-627)DOAJ008124957 (DE-599)DOAJdb3c343ea7604ebc86dbbd75e7d4b887 DE-627 ger DE-627 rakwb eng rus ukr Q1-390 Ya.I. Grushka verfasserin aut Necessary and sufficient condition for the existence of the one-point time on an oriented set 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. changeable sets time ordered sets oriented sets Science (General) In Доповiдi Нацiональної академiї наук України Publishing House "Akademperiodyka", 2019 8(2019), Seite 9-15 (DE-627)66080784X (DE-600)2609380-7 2518153X nnns volume:8 year:2019 pages:9-15 https://doi.org/10.15407/dopovidi2019.08.009 kostenfrei https://doaj.org/article/db3c343ea7604ebc86dbbd75e7d4b887 kostenfrei http://www.dopovidi-nanu.org.ua/en/archive/2019/8/2 kostenfrei https://doaj.org/toc/1025-6415 Journal toc kostenfrei https://doaj.org/toc/2518-153X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 8 2019 9-15
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allfieldsGer	10.15407/dopovidi2019.08.009 doi (DE-627)DOAJ008124957 (DE-599)DOAJdb3c343ea7604ebc86dbbd75e7d4b887 DE-627 ger DE-627 rakwb eng rus ukr Q1-390 Ya.I. Grushka verfasserin aut Necessary and sufficient condition for the existence of the one-point time on an oriented set 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. changeable sets time ordered sets oriented sets Science (General) In Доповiдi Нацiональної академiї наук України Publishing House "Akademperiodyka", 2019 8(2019), Seite 9-15 (DE-627)66080784X (DE-600)2609380-7 2518153X nnns volume:8 year:2019 pages:9-15 https://doi.org/10.15407/dopovidi2019.08.009 kostenfrei https://doaj.org/article/db3c343ea7604ebc86dbbd75e7d4b887 kostenfrei http://www.dopovidi-nanu.org.ua/en/archive/2019/8/2 kostenfrei https://doaj.org/toc/1025-6415 Journal toc kostenfrei https://doaj.org/toc/2518-153X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 8 2019 9-15
allfieldsSound	10.15407/dopovidi2019.08.009 doi (DE-627)DOAJ008124957 (DE-599)DOAJdb3c343ea7604ebc86dbbd75e7d4b887 DE-627 ger DE-627 rakwb eng rus ukr Q1-390 Ya.I. Grushka verfasserin aut Necessary and sufficient condition for the existence of the one-point time on an oriented set 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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. changeable sets time ordered sets oriented sets Science (General) In Доповiдi Нацiональної академiї наук України Publishing House "Akademperiodyka", 2019 8(2019), Seite 9-15 (DE-627)66080784X (DE-600)2609380-7 2518153X nnns volume:8 year:2019 pages:9-15 https://doi.org/10.15407/dopovidi2019.08.009 kostenfrei https://doaj.org/article/db3c343ea7604ebc86dbbd75e7d4b887 kostenfrei http://www.dopovidi-nanu.org.ua/en/archive/2019/8/2 kostenfrei https://doaj.org/toc/1025-6415 Journal toc kostenfrei https://doaj.org/toc/2518-153X Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 8 2019 9-15
language	English Russian Ukrainian
source	In Доповiдi Нацiональної академiї наук України 8(2019), Seite 9-15 volume:8 year:2019 pages:9-15
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author	Ya.I. Grushka
spellingShingle	Ya.I. Grushka misc Q1-390 misc changeable sets misc time misc ordered sets misc oriented sets misc Science (General) Necessary and sufficient condition for the existence of the one-point time on an oriented set
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topic_title	Q1-390 Necessary and sufficient condition for the existence of the one-point time on an oriented set changeable sets time ordered sets oriented sets
topic	misc Q1-390 misc changeable sets misc time misc ordered sets misc oriented sets misc Science (General)
topic_unstemmed	misc Q1-390 misc changeable sets misc time misc ordered sets misc oriented sets misc Science (General)
topic_browse	misc Q1-390 misc changeable sets misc time misc ordered sets misc oriented sets misc Science (General)
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title	Necessary and sufficient condition for the existence of the one-point time on an oriented set
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title_full	Necessary and sufficient condition for the existence of the one-point time on an oriented set
author_sort	Ya.I. Grushka
journal	Доповiдi Нацiональної академiї наук України
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author-letter	Ya.I. Grushka
doi_str_mv	10.15407/dopovidi2019.08.009
title_sort	necessary and sufficient condition for the existence of the one-point time on an oriented set
callnumber	Q1-390
title_auth	Necessary and sufficient condition for the existence of the one-point time on an oriented set
abstract	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.
abstractGer	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.
abstract_unstemmed	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.
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title_short	Necessary and sufficient condition for the existence of the one-point time on an oriented set
url	https://doi.org/10.15407/dopovidi2019.08.009 https://doaj.org/article/db3c343ea7604ebc86dbbd75e7d4b887 http://www.dopovidi-nanu.org.ua/en/archive/2019/8/2 https://doaj.org/toc/1025-6415 https://doaj.org/toc/2518-153X
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up_date	2024-07-03T16:06:54.290Z
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