Serial relation: Difference between revisions
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{{Short description|Relation that relates every element to some element}} | |||
In [[Set theory|set theory]] a '''serial relation''' is a [[Homogeneous relation|homogeneous relation]] expressing the connection of an element of a [[Sequence|sequence]] to the following element. The [[Successor function|successor function]] used by [[Biography:Giuseppe Peano|Peano]] to define [[Natural number|natural number]]s is the prototype for a serial relation. | In [[Set theory|set theory]] a '''serial relation''' is a [[Homogeneous relation|homogeneous relation]] expressing the connection of an element of a [[Sequence|sequence]] to the following element. The [[Successor function|successor function]] used by [[Biography:Giuseppe Peano|Peano]] to define [[Natural number|natural number]]s is the prototype for a serial relation. | ||
[[Biography:Bertrand Russell|Bertrand Russell]] used serial relations in ''[[The Principles of Mathematics]]''<ref name=":0">{{Cite book |author=Russell, Bertrand |url=http://worldcat.org/oclc/1203009858 |title=Principles of mathematics |isbn=978-1-136-76573-5 |oclc=1203009858}}</ref> (1903) as he explored the foundations of [[Order theory|order theory]] and its applications. The term ''serial relation'' was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: [[Connected relation|connectedness]], irreflexivity, and [[Transitive relation|transitivity]].<ref>B. A. Bernstein (1926) [https://www.ams.org/journals/bull/1926-32-05/S0002-9904-1926-04256-7/S0002-9904-1926-04256-7.pdf "On the Serial Relations in Boolean Algebras"], [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] 32(5): 523,524</ref> | |||
[[Biography:Bertrand Russell|Bertrand Russell]] used serial relations in ''[[The Principles of Mathematics]]''<ref name=":0">{{Cite book |author=Russell, Bertrand |url=http://worldcat.org/oclc/1203009858 |title=Principles of mathematics |date=25 February 2020 |publisher=Routledge |isbn=978-1-136-76573-5 |oclc=1203009858}}</ref> (1903) as he explored the foundations of [[Order theory|order theory]] and its applications. The term ''serial relation'' was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: [[Connected relation|connectedness]], irreflexivity, and [[Transitive relation|transitivity]].<ref>B. A. Bernstein (1926) [https://www.ams.org/journals/bull/1926-32-05/S0002-9904-1926-04256-7/S0002-9904-1926-04256-7.pdf "On the Serial Relations in Boolean Algebras"], [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] 32(5): 523,524</ref> | |||
A serial relation ''R'' is an endorelation on a set ''U''. As stated by Russell, <math>\forall x \exists y \ xRy ,</math> where the universal and existential quantifiers refer to ''U''. In contemporary language of [[Relation (mathematics)|relation]]s, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest. | A serial relation ''R'' is an endorelation on a set ''U''. As stated by Russell, <math>\forall x \exists y \ xRy ,</math> where the universal and existential quantifiers refer to ''U''. In contemporary language of [[Relation (mathematics)|relation]]s, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest. | ||
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For a relation ''R'', let {{mset|''y'': ''xRy''}} denote the "successor neighborhood" of ''x''. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an '''inverse serial''' relation is a relation in which every element has non-empty "predecessor neighborhood".<ref name="Yao2005">{{Cite book | doi = 10.1007/978-3-540-27778-1_15| chapter = Semantics of Fuzzy Sets in Rough Set Theory| title = Transactions on Rough Sets II| volume = 3135| pages = 309| series = [[Lecture Notes in Computer Science]]| year = 2004| last1 = Yao | first1 = Y. | isbn = 978-3-540-23990-1}}</ref> | For a relation ''R'', let {{mset|''y'': ''xRy''}} denote the "successor neighborhood" of ''x''. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an '''inverse serial''' relation is a relation in which every element has non-empty "predecessor neighborhood".<ref name="Yao2005">{{Cite book | doi = 10.1007/978-3-540-27778-1_15| chapter = Semantics of Fuzzy Sets in Rough Set Theory| title = Transactions on Rough Sets II| volume = 3135| pages = 309| series = [[Lecture Notes in Computer Science]]| year = 2004| last1 = Yao | first1 = Y. | isbn = 978-3-540-23990-1}}</ref> | ||
In [[Philosophy:Normal modal logic|normal modal logic]], the extension of fundamental axiom set '''K''' by the serial property results in axiom set '''D'''.<ref>[[Biography:James Garson|James Garson]] (2013) ''Modal Logic for Philosophers'', chapter 11: Relationships between modal logics, figure 11.1 page 220, | In [[Philosophy:Normal modal logic|normal modal logic]], the extension of fundamental axiom set '''K''' by the serial property results in axiom set '''D'''.<ref>[[Biography:James Garson|James Garson]] (2013) ''Modal Logic for Philosophers'', chapter 11: Relationships between modal logics, figure 11.1 page 220, Cambridge University Press {{doi|10.1017/CBO97811393421117.014}}</ref> | ||
== Russell's series == | == Russell's series == | ||
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=== Stretch === | === Stretch === | ||
Russell adopted the term ''stretch'' from [[Biography:Alexius Meinong|Meinong]], who had contributed to the theory of distance.<ref>[[Biography:Alexius Meinong|Alexius Meinong]] (1896) ''Uber die Bedeutung der Weberische Gesetze''</ref> Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."<ref name=":0" />{{Rp|page=181}} To explain Meinong, Russell refers to the [[Cayley–Klein metric]], which uses stretch coordinates in anharmonic ratios which determine distance by using logarithm.<ref name=":0" />{{rp|page=255}}<ref>Russell (1897) ''An Essay on the Foundations of Geometry''</ref> | Russell adopted the term ''stretch'' from [[Biography:Alexius Meinong|Meinong]], who had contributed to the theory of distance.<ref>[[Biography:Alexius Meinong|Alexius Meinong]] (1896) ''Uber die Bedeutung der Weberische Gesetze''</ref> Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."<ref name=":0" />{{Rp|page=181}} To explain Meinong, Russell refers to the [[Cayley–Klein metric]], which uses stretch coordinates in [[Cross ratio|anharmonic ratios]] which determine distance by using logarithm.<ref name=":0" />{{rp|page=255}}<ref>Russell (1897) ''An Essay on the Foundations of Geometry''</ref> | ||
== References == | == References == | ||
Latest revision as of 00:34, 15 April 2026
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
Bertrand Russell used serial relations in The Principles of Mathematics[1] (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.[2]
A serial relation R is an endorelation on a set U. As stated by Russell, where the universal and existential quantifiers refer to U. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.
For a relation R, let {y: xRy} denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".[3]
In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.[4]
Russell's series
Relations are used to develop series in The Principles of Mathematics. The prototype is Peano's successor function as a one-one relation on the natural numbers. Russell's series may be finite or generated by a relation giving cyclic order. In that case, the point-pair separation relation is used for description. To define a progression, he requires the generating relation to be a connected relation. Then ordinal numbers are derived from progressions, the finite ones are finite ordinals.[1]: Chapter 28: Progressions and ordinal numbers Distinguishing open and closed series[1]: 234 results in four total orders: finite, one end, no end and open, and no end and closed.[1]: 202
Contrary to other writers, Russell admits negative ordinals. For motivation, consider the scales of measurement using scientific notation, where a power of ten represents a decade of measure. Informally, this parameter corresponds to orders of magnitude used to quantify physical units. The parameter takes on negative as well as positive values.
Stretch
Russell adopted the term stretch from Meinong, who had contributed to the theory of distance.[5] Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole."[1]: 181 To explain Meinong, Russell refers to the Cayley–Klein metric, which uses stretch coordinates in anharmonic ratios which determine distance by using logarithm.[1]: 255 [6]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Russell, Bertrand (25 February 2020). Principles of mathematics. Routledge. ISBN 978-1-136-76573-5. OCLC 1203009858. http://worldcat.org/oclc/1203009858.
- ↑ B. A. Bernstein (1926) "On the Serial Relations in Boolean Algebras", Bulletin of the American Mathematical Society 32(5): 523,524
- ↑ Yao, Y. (2004). "Semantics of Fuzzy Sets in Rough Set Theory". Transactions on Rough Sets II. Lecture Notes in Computer Science. 3135. pp. 309. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.
- ↑ James Garson (2013) Modal Logic for Philosophers, chapter 11: Relationships between modal logics, figure 11.1 page 220, Cambridge University Press doi:10.1017/CBO97811393421117.014
- ↑ Alexius Meinong (1896) Uber die Bedeutung der Weberische Gesetze
- ↑ Russell (1897) An Essay on the Foundations of Geometry
External links
- Jing Tao Yao and Davide Ciucci and Yan Zhang (2015). "Generalized Rough Sets". in Janusz Kacprzyk and Witold Pedrycz. Handbook of Computational Intelligence. Springer. pp. 413–424. ISBN 9783662435052. https://books.google.com/books?id=gLS4CQAAQBAJ&pg=PA416. Here: page 416.
- Yao, Y.Y.; Wong, S.K.M. (1995). "Generalization of rough sets using relationships between attribute values". Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33. http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf..
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