Talk:Serial relation

Russell

The following citation agrees on the phrase "serial relation" but differs from the concept of this article, so it was removed:

Bertrand Russel referred to serial relations in his Principles of Mathematics when he introduced ordinal numbers. He writes

The finite ordinals may be conceived as types of series: for example, the ordinal number n may be taken to mean "a serial relation of n terms;" or, in popular language, n terms in a row. This is an ordinal notion, distinct from "nth", and logically prior to it. In this sense, n is the name of a class of serial relations.<ref B. Russell (1903) The Principles of Mathematics, page 454, §290 Transfinite Ordinals, link from University of Massachusetts /ref>

He also describes a relation formed by a sum of serial relations.<ref Principles of Mathematics, page 468 /ref>

This article pertains to relatons R such that ∀x∃y xRy. — Rgdboer (talk) 21:16, 7 April 2018 (UTC)

"Domain" as used in the definition

The article currently defines a serial relation as

a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y  x R y)".

However, Binary relation#Formal definition defines the domain of a binary relation R as

the set of all x such that x R y for at least one y,

making the definition of a serial relation as it is currently worded vacuous or at least somewhat inconsistent with the definitions given at Binary relation (I think). – Tea2min (talk) 07:07, 24 May 2018 (UTC)

Excellent point, according to the "formal definition" every binary relation is serial. Three set theory refs are given in the formal def; will see if they concur. Meanwhile see Talk:Binary relation#Graph theory for current discussion of the formal def. — Rgdboer (talk) 22:42, 24 May 2018 (UTC)

All three references adhere to the domain, range, field terminology. Evidently a binary relation is serial on its domain. However, a binary relation may be contained in another that has a larger domain where the first relation is not serial. — Rgdboer (talk) 23:09, 24 May 2018 (UTC)

Complete relation

Complete relation currently redirects to Serial relation, but the term "complete relation" is not mentioned there at all. Complete relation used to redirect to Total relation before that was turned into a redirect to Serial relation, and Total relation used to mention "complete relation" as a synonym (see old version of "Total relation"). – Tea2min (talk) 13:52, 30 May 2018 (UTC)

Redirect has been changed to Connex relation where it is mentioned, but with a tag asking for a reference. Rgdboer (talk) 23:43, 17 July 2019 (UTC)

Totality

"Serial relation" is little used while "Total relation" has come to the fore, especially in contrast to partial function. The concept of Totality is important in algebra as seen in the template on group-like structures. Thus it appears this article should be moved to Total relation (a redirect currently brings readers here). Does anyone object to such a Move? — Rgdboer (talk) 23:53, 17 July 2019 (UTC)

I guess you are right about usage conventions. Nevertheless I'd prefer "serial" since it avoids the possibility of confusion. I marked with {{citation needed}} the claims about "total" used for "serial" as well as for "connex". If no citations show up within reasonable time, I'd possibly change my mind. - Jochen Burghardt (talk) 08:37, 18 July 2019 (UTC)

References for Total have been supplied. Since the confusion with Connex should not be perpetuated, and a reference is not at hand, the following was removed:

The term total relation has also been used to designate a connex relation, i.e., a homogeneous binary relation R for which either xRy or yRx holds for any pair (x,y).

Our efforts are coordinated to assist readers to find current terminology. — Rgdboer (talk) 20:58, 19 July 2019 (UTC)

Serial relation is a generic term referring to various relation Russell used in 1903 to develop order theory through series. Today the topic was added as "Russell's series". Currently total relation is a disambiguation claiming that Connected relationis an alternative article. Yet the only reference given is Aliprantis and Border (2007) Infinite Dimensional Analysis. The Introduction and Characterization sections are more commonly called Total. Rgdboer (talk) 23:41, 17 September 2021 (UTC)

The links to this article are being pruned and sent to Total relation, which is the commonly used term. Serial relation is a concept developed by B. Russell to discuss series, both infinite and finite. Rgdboer (talk) 03:39, 13 April 2022 (UTC)

@Rgdboer: "Total" is often used for "connex" in the subfield of ordering relations, like in "Let R be a partial order. We prove that ... - If R is total, we can even prove that ...". The "total" there means "connex", i.e. (partial order) ∧ (connex relation) ↔ (total order). People who are familiar with this subfield are likely to be confused when "total" is used for "serial". Therefore, I strongly recommend to somehow separate both meanings by using distinct names. If you insist that "serial" should not be used, what about "left total"? - Jochen Burghardt (talk) 07:43, 13 April 2022 (UTC)

Well “total relation” and “total order” might be thought total for the same reason, but the adjective “total” has different meanings in these phrases. References are being assembled and note taken of the citation requested at homogeneous relation where “connected” is described as possibly being “total”. Rgdboer (talk) 03:16, 14 April 2022 (UTC)

@Rgdboer: "but the adjective “total” has different meanings in these phrases" That's exactly my point. We should try to avoid such ambiguous words, in order to prevent confusion. - Jochen Burghardt (talk) 08:45, 14 April 2022 (UTC)

An order may be total as a relation, but not be a total order!! This difficulty in descriptive language suggests these notes: heterogeneous relations may be total, but orders are endorelations, and the connectedness axiom needs that context. General readers may be interested in relations per se, or orders, and the texts of articles ought to direct them accordingly. A redirect total relation (order) for Connected relation provides an expedient for order theory contexts.Rgdboer (talk) 03:51, 17 April 2022 (UTC)

Proposed merge of Total relation into Serial relation

Seems to be a duplicate article. 1234qwer1234qwer4 17:32, 4 May 2022 (UTC)

Totality is only a property of a serial relation, which is a homogeneous relation. A total relation may be heterogeneous. Rgdboer (talk) 01:26, 7 May 2022 (UTC)