855:(I posed the initial question.) Thanks, this is much better. One often does not read the whole article, and while everyone thinks to know what "smallest" means and thus gets confused, the word "initial" does not have this problem. ... Also, I apologise if I didn't do something quite properly; it's been ages since I last edited Knowledge (not the English one) and I am therefore familiar with neither the written not the unwritten rules. But I guess when one only edits the "talk", one cannot do that much harm. Thanks once more, cheers!
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totally ordered set, there being no element of ∅ capable of serving as a bound. On the other hand, if we view these sets as proper subsets of some other totally-ordered superset, then ∅ is not an unbounded set, being bounded by every element of its superset. But in this case whether the natural numbers or integers are bounded depends upon the nature of the total order of their superset. For example if the natural numbers are viewed as an initial segment of some larger ordinal, then they are bounded above.
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formula without a name. Grätzer ("General
Lattice Theory", Birkhäuser, 2003) uses on p.1 the name "linearity" property, and on p.2 "chain" and all of Birkhoff's synonyms. A calculus textbook, Heuser ("Lehrbuch der Analysis 1", B.G.Teubner, 1980, I don't have English ones available) defines on p.36 the notion of
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I have rewritten the section. I have not changed the references, and I left to others to add further references and/or to find a better place, if needed, for the existing ones. I think that this version is much better than the previous one, but feel free for improving it further (and fix my mistakes,
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I expected the section "Strict and non-strict total orders" to contain a definition of a non-strict total order. It currently only defines only strict total orders and then discusses relationship between strict and non-strict total orders. Including a definition would help readers' understandings of
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The article states "The rational numbers comprise the smallest totally ordered set which is dense in the real numbers.". I don't think that's true. Smallest in which sense? I think that e.g. rational numbers whose denominators are powers of two are one possible "smaller" dense set (and every subset
677:). So a total order is never a strict total order. The article explains in detail the relationship between the two concepts, namely that there are equivalent (a total order defines a strict total order, a strict total order defines a total order, and these two operations are inverses one to other).
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For example, Birkhoff ("Lattice Theory", AMS Colloquium
Publications vol. 25, 1967) defines on p.1 "partial ordered set", naming all three involved properties ("Reflexive", "Antisymmetry", "Transitivity"), on p.2 he defines "simply ordered set" = "totally ordered set" = "chain", showing the connex
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Minimally the the word "non-empty" needs to be inserted, which I will do shortly. Beyond that, either the statements in this article should to be revised to make it clear that the sets are being viewed as subsets of themselves, or if it is commonplace in current mathematical discourse to refer to
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property. Halmos ("Naive
Mengenlehre", Vandenhoeck&Ruprecht, 1976; German translation of "Naive Set Theory", Nostrand, 1968, I have only the German book) defines in ch.14 on p.72 "reflexive", "anti-symmetric", "transitive" relations, and "total" = "simple" = "linear" "orders", again without
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only talks about upper and lower bounds of subsets of an ordered set. If we interpret the above remarks about the natural numbers and integers as statements about these sets as subsets of themselves, then the remarks are not correct. ∅ (viewed as a subset of itself) is the smallest unbounded
570:). However, the adjective "linear" is not a good alternative, since it is associated with vector spaces, and "simple" cannot be associated to anything. I believe I saw "fully ordered set" (as opposed to "partially ordered set") somewhere, but "full" isn't a good adjective either. -
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I have some difficulties for rewriting the section, because it needs to report common usages of a concept that is rarely formally defined. As far as I know, "a chain" refers generally to a set of subsets of a given set, which is totally ordered by inclusion. For example, the
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A problem with "connex" is that it is not an adjective, thus often requiring clumsy grammatical constructions in enumerations (e.g. "A relation is trichotomous if, and only if, it is irreflexive, asymmetric, and a semi-connex relation." at
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Davey.Priestley.1990 define "chain" as in 1. (p.3), but use it mostly in the sense of 1.1. (Zorn: p.100, chain-completeness p.5); they use "sequence", not "chain" in the definition of ACC and DCC (p.38), and use 1.1. for Zorn's lemma
823:"Smallest" is defined a few lines above, but is confusing as the definition does not excludes the existence of an strict subset that is order isomorphic (for example, the even integers are order isomorphic to the integers).
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is stated for any ordered set, but most of its applications are for sets (of some specific type) ordered by inclusion. So, the section must mention three main contexts where the term "chain" is used: for stating and using
491:). I am not really happy with these changes since every source I have ever seen that defined the term "total order" used the word "totality" to denote the property that every element is related to every other element. –
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of a commutative ring is the maximal length of chains of prime ideals. It seems that the use of "chain" for other partial orders than inclusion results generally from a generalization of this case. For example,
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of elements of an ordered set such that each element of the sequence is either greater (ascending chain) or smaller (descending chain) than the preceding one. In other words, a chain is the image of a
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When viewing the article on mobile the binary relations table shows first, fully expanded. I’m not sure how best to fix this (and I am on mobile) but it seems an unhelpful way to begin new readers.
1072:; for finite chains that are used for many notions of dimensions (the dimension of a vector space the the maximal length of chains of linear subspaces), and also in graph theory as a synonym of
558:), I think the rewording is ok. However, we should add warning sentences or footnotes about the two meanings of "totality" at appropriate places. I tried to phrase such a warning in the lead of
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944:). Since ascending and descending chains are introduced below as special cases of chains, probably this can be cleaned up simply by removing everything between "chain" and "is often used"?
980:, "chain" is used as a synonym for a totally ordered subset. Also, in the discussion of ascending chain conditions or descending chain conditions, chains are not limited to sequences. —-
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542:). After I saw the rewording, I glanced at some textbooks, and found that "total(ly) order(ed set)" is used often, but -to my surprise- "totailty property" and "total relation" is not.
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Therefore, and since recent publications appear to use "connex" more often (this word didn't appear at all in the above books; a few recent publications that use it are listed at
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Gratzer.2003 defines "chain" as in 1., and "chain in a PO" as in 1.1. but nonempty (both p.2); he defines "increasing chain" on the fly as in 1.2. (p.15, exercise about ACC).
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is equivalent in the classical logic, but is stronger in the intuitionistic logic. (So one who uses it is advised to choose the "Strict Total Order" in one's definitions.)
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I intend to rewrite this section. However I am not sure that this article is the right place for defining chains. So, I wait for further input before editing this section.
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is reliable but it defines a chain as a totally ordered subset; so any subset (even empty??) of R is a chain. I doubt we know whether the empty set is a chain or not. —-
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Being not about the subject of this article, this table does not belong to the lead. I have moved it to the section "Related structures, where it is more appropriate.
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You are right that "chain" as a straightforward synonym for "totally ordered set" is uncommon, but there is a reference given in the lead; here's another:
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This is the way I intend to rewrite the section. However I may miss some important things, and a feedback on the above summary would be welcome.
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An occurence of "strict" was omitted. Otherwise, the definition of a (non strict) total order is given in the preceding section
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The problem is that "total relation" can be used to describe two different and unrelated properties, viz.
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So, I have replaced "smallest" by "initial", as the concept is close (although weaker) to the concept of
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function of the natural numbers (or of an initial interval of the natural numbers) into an ordered set.
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I added the Fraïssé reference, and removed the ((dubious)) tag; this shouldn't interfer too much with
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on 12 February 2020. For the contribution history and old versions of the redirected page, please see
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p. 35. The use of "chain" for "totally ordered subset of a partially ordered set" is very common (eg
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would be a chain. I never heard such an assertion. As far as I know, a "chain" generally refers to a
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Knowledge. If you would like to participate, please visit the project page, where you can join
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This also shows that an author needn't stick with a single node in the hierarchy. -
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uses D.Lazard's notion of "chain", viz. a sequence of elements of an ordered set. -
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talks about a countable chain, which suggests there can be a uncountable one :) --
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The integers comprise the smallest totally ordered set with neither an upper nor a
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A total order is currently being defined as follows: "That is, a total order is a
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From your description, I'd suggest the following hierachy of meanings of "chain":
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I think I'm sure I've seen one, but can't think of an example just right now.
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Neither did I. Do you have another suggestion about how to avoid confusion? -
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I never heard or saw the word "connex" before it was added to this article.
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Either way, the statements as currently written are not universally true.
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The natural numbers comprise the smallest totally ordered set with no
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I would not oppose to remove all mentions of "smallest / initial".
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I think it is possible to add a remark that the approach
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This is redundant – 1. follows from 4. if you substitute
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Example of total order that is not a strict total order?
988:) 07:04, 24 April 2021 (UTC) I don’t know if this note
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of the reals is clearly totally ordered). Am I right?
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The unique order on the empty set, ∅, is a total order.
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1251:(2nd ed.). Basel: Birkhäuser.
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890:{\displaystyle \mathbb {R} }
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925:Fraïssé, R. (2011-08-18).
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1257:3-7643-6996-5
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123:Find sources:
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110:Verifiability
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52:Learn to edit
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1062:Zorn's lemma
1057:Zorn's lemma
978:Zorn's lemma
962:Richard Zach
946:Richard Zach
929:. Elsevier.
926:
907:
874:
857:46.13.147.72
809:46.13.147.72
803:— Preceding
799:
778:Georgydunaev
772:— Preceding
769:
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740:— Preceding
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271:Mid‑priority
249:WikiProjects
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19:This is the
716:lower bound
709:upper bound
481:instead of
475:Total order
473:), and now
393:its history
388:Total order
321:Mathematics
312:mathematics
268:Mathematics
148:free images
31:not a forum
25:Total order
1853:Categories
1540:transitive
1204:References
1119:order type
548:trichotomy
382:page were
1455:reflexive
1186:if any).
590:JRSpriggs
429:Archive 1
237:is rated
88:if needed
71:Be polite
21:talk page
1820:D.Lazard
1805:Bradknox
1335:on some
1294:D.Lazard
1234:89009753
1188:D.Lazard
1161:(p.100).
1081:D.Lazard
1029:D.Lazard
910:D.Lazard
899:sequence
843:D.Lazard
805:unsigned
786:contribs
774:unsigned
742:unsigned
679:D.Lazard
456:Totality
417:Archives
186:Archives
56:get help
29:This is
27:article.
493:Tea2min
348:on the
239:B-class
154:WP refs
142:scholar
1759:Joriki
1139:1.3.1.
1064:; for
522:) and
384:merged
245:scale.
126:Google
1597:then
1512:then
386:into
226:This
169:JSTOR
130:books
84:Seek
1838:talk
1824:talk
1809:talk
1788:talk
1779:Done
1763:talk
1755:").
1709:for
1571:and
1486:and
1384:and
1298:talk
1282:talk
1278:HTGS
1253:ISBN
1231:LCCN
1222:ISBN
1192:talk
1174:talk
1133:1.3.
1115:1.2.
1109:1.1.
1085:talk
1041:talk
1015:talk
1011:Taku
997:talk
993:Taku
986:talk
982:Taku
966:talk
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914:talk
861:talk
847:talk
813:talk
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683:talk
632:talk
612:talk
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576:talk
497:talk
488:diff
470:diff
162:FENS
136:news
73:and
1655:or
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1460:If
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540:yRx
536:xRy
520:xRy
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176:TWL
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