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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 [[Giuseppe Peano|Peano]] to define [[natural number]]s is the prototype for a serial relation.

[[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]] 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"], [[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.

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 [[normal modal logic]], the extension of fundamental axiom set '''K''' by the serial property results in axiom set '''D'''.<ref>[[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 ==
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 number]]s. 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 number]]s are derived from progressions, the finite ones are finite ordinals.<ref name=":0" />{{Rp|location=Chapter 28: Progressions and ordinal numbers}} Distinguishing open and closed series<ref name=":0" />{{Rp|page=234}} results in four total orders: finite, one end, no end and open, and no end and closed.<ref name=":0" />{{Rp|page=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 (log scale)|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 [[Alexius Meinong|Meinong]], who had contributed to the theory of distance.<ref>[[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 ==
{{Reflist}}

== External links ==
* {{cite book | chapter-url=https://books.google.com/books?id=gLS4CQAAQBAJ&pg=PA416 | author=Jing Tao Yao and Davide Ciucci and Yan Zhang | contribution=Generalized Rough Sets | pages=413–424 | isbn=9783662435052 | editor=Janusz Kacprzyk and Witold Pedrycz | title=Handbook of Computational Intelligence | publisher=Springer | year=2015 }} Here: page 416.
* {{cite journal|last = Yao|first = Y.Y.|author2=Wong, S.K.M.|title = Generalization of rough sets using relationships between attribute values|journal = Proceedings of the 2nd Annual Joint Conference on Information Sciences|year = 1995|pages = 30–33|url = http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf}}.

[[Category:Properties of binary relations]]
[[Category:Order theory]]

Serial relation - Revision history

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