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coordinates where they disagree is a nonempty set, has a least element, compare at that coordinate.) In particular, you could do this for a family of well-ordered sets, but in this case, the dictionary order might not be well-ordered. (Consider the dictionary order on 2, 2, 2, 2, ..., where the index ordinal is w. Then 1, 1, 1, 1, 1, ...; 1, 0, 1, 1, 1 ; ..., 1, 0, 0, 1, 1, ...; etc. is a strictly decreasing sequence in 2, 2, 2, ..., so it can't be well-ordered.)
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1132:(word over an alphabet) and that this case is not even described. Moreover the other variants of the lexicographical order may easily be described by reducing them to the case of sequences, and this gives a description which is much easier to understand (except, maybe, for specialists of abstract set theory) than that which is presently given.
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The set of words of finite length, even over a finite alphabet, is not well ordered by the lexicographic order. Consider words over the alphabet {0, 1} (with 0 < 1), and consider the subset S of all words whose last letter is 1. If the lexicographic order were a well-order, S should have a least
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Are the image labels accurate? The 2 labels above the very left grids of red and white squares seem backwards. The labels above the circled numbers all make sense, and on the right side of the image, the labels for the red and white squares match the labels above the circled numbers, but on the left
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Nevertheless, "dictionary order" is standard mathematical terminology, regardless of whether it completely conforms to the ordinary use of the term. BTW, I see no reason why the number of sets cannot be generalised to range over any ordinal number, (e.g. here it is given for a finite ordinal, but the
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This is a terribly confusing set of explanations of a relatively simple idea. Mathematicians are increasingly using terminology and notation that are more complex than the ideas they are trying to describe. Mathematics is becoming swamped with people more concerned with definitions and notation than
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which are not defined in
Knowledge nor in the source. These have not any evident meaning as there are more than two orders that are involved here (the order being defined, the order used to convert subsets into sequences, the order for reading the sequences, the order for comparing elements, and the
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Does this whole thing fall apart if they are mostly ordered, or completely unordered, or what? And what on earth is this about a
Cartesian product? What does multiplication have to do with putting a series in order, and how can two sets be multiplied, partially ordered or otherwise? Numerous puzzles
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I think the red ordering (binary vectors) is correct or incorrect whether we consider that the most significant bit is on the left or right. If we consider that the least significant bit is on the left, then the red and the blue ordering should be the same. Can someone confirm or infirm ? If so, it
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Yes, it looks like there's something wrong. On the right-hand side as well, according to my understanding. The various instances of the word "Rev" seem to be misplaced. Nothing we can immediately do, since the image is not editable, but unless someone can enlighten us as to its correctness, I think
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Second the motion... think of me as an intelligent but uninformed reader. As it stands now it says: "Given two partially ordered sets A and B, the lexicographical order on the
Cartesian product A × B is defined as (a,b) ≤ (a′,b′) if and only if a < a′ or (a = a′ and b ≤ b′)." I feel like a bunch
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OK, I see, I guess it's correct. But for this to make any sense, it all needs to be explained in the article. There seems to be nothing in the article about binary vectors, or how they might "correspond" to subsets (and in fact I don't think it's strictly subsets that are being ordered here, but
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This could be misinterpreted as implying a_1!=b_1 due to the comma appearing before "which," when the intention was clearly to define i to be the least i such that a_i!=b_i. It seems like bad practice to place the definition of a variable inside what one might interpret as an appositive phrase,
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I made it like that for a purpose. You can see that the colors in the arrangements of 20x3 balls on the left and on the right are horizontally reflected, although the numbers are not. So we can see how colex order is related to lex order. The similarity between lex and colex order is not easily
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sets, indexed by an ordinal number, then you could define the dictionary ordering on that family, and it would again turn out to be a totally ordered set. (Define: if two elements aren't equal, then they disagree in a coordinate, these coordinates are indexed by an ordinal, so the set of all
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Ah, I think I see where I screwing up. It is true (I checked a set theory book) that the product of two ordinals is the dictionary ordering on the cartesian product, with the order of factors reversed. And, it also seems true (I haven't gone through the details) that if you have a family of
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Isn't this just ordinal multiplication, with the order of A and B reversed? If this is the case, is it possible to define ordinal multiplication where the factors themselves are indexed by an ordinal? I've seen cardinal multiplication defined for indexed families, but not ordinal.
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says, "The lexicographic ordering of strings is the same as the familiar dictionary ordering, except that shorter strings precede longer strings. Thus the lexicographic ordering of all strings over the alphabet {0,1} is {empty,0,1,00,01,10,11,000,...}." That would seem to be the
450:... is an infinite decreasing sequence of elements. In general strings of fixed length do preserver the well-ordering property but by using the 'space padding' method described in the article to extend the definition to the LR-lex order the property no longer holds.
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I'm taking classes again and need a quick reference to the word. Found extensive discussion that was so much more useful and enlightening on the subject. Now I can go back to class with a much better understanding of, and ability within the subject area. Thank you
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Why did we have a C++ string comparison function here? It is, at most, marginally related to the topic of this article. Moreover, why case insensitive? Lexicographic orders have nothing to do (are ortogonal) to being case (in)sensitive. Since, anyway,
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There is also a wreath product of orderings on sets that preserves the well-ordering property but it is a bit involved to explain here. Maybe at some point in the future I'll take a shot at rewriting this page to include these definitions.
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The current text in "Definition and
Motivation" replaced a significantly more abstruse version. If you have suggestions for improvement, please add them below. And please sign your comments by appending four tildes (~~~~) at the end. --
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ordered triples). I also agree with the proposal to merge the article on colex order with this one. And there seems to be inconsistency in the definition of reverse lex order as given in the article and as used here. Lots to sort out.
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of knives are being tossed at me to catch. A, B, got 'em. Now here's a and b, what are lower case a,b? part of sets A or B or something else? a and b are not defined! And what is the significance of A and B being
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This should be mentioned in the article itself, no other source mentions that strings of finite length is not well ordered on lexicographic order, and the statement in the article itself is misleading by itself.
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word w. But for any putative least word w in S, you can change the last letter from 1 to 01, yielding another word in S that precedes w in the lexicographic order. Thus there can be no least element of S. --
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There should be a more accessible introduction, which gives an example that older elementary school children (and for that matter high school students and non-mathematically inclined adults) can understand. --
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An important property of the lexicographical order is that it preserves well-orders, that is, if A and B are well-ordered sets, then the product set A × B with the lexicographical order is also well-ordered.
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here, all of which require research before I can solve them, and only then can I learn what I came here to learn. I suggest that if you can't express it clearly you might just not understand it yourself.
446:.) The standard dictionary order is what (Sims 94, Computation With Finitely Presented Groups) calls the left-right-lexicographic order. This is not a well-order, since if a < b then b : -->
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In the last sentence of the "Motivation and definition" section, when it says "of a fixed length", this seems a bit trivial and silly, maybe what it meant to say is "of finite length"?
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shown, and this is one way to do it. However: The actual information are the white numbers. The bluish colors are just a background to make the pattern comprehensible.
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634:) which describes it. Let us expand it! I'll start by adding the alternate names Length-plus-lexicographic and Radix order - that's the name I knew for it. --
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The "Definition" section would be a good place to introduce the term. More material could be put in the "Reverse lexicographic order" section if necessary.
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The article says (paraphrased) the word A comes before B "if and only if the first a_i, which is different from b_i, comes before b_i in the alphabet.
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I have largely rewritten the section "Motivation and definition" in response to the "clarify" tag placed there in March, 2020. Hope it helps. --
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There is a problem here with how the ordering is extended to products of different lengths. (For simplicity I'll assume we have a fixed alphabet
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can ever be more than a dictionary definition; anything of encyclopedic value that can be said in one article can also be said in the other.
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I added a paragraph on the ordering of strings, in which I also mention what you call Length-plus-lexicographic ordering; I found a stub (
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But why does it say "Rev Lex" above the top left chart, and "Lex" above the bottom left one? Shouldn't these be the other way round?
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As it is, the article is highly confusing and misleading. The main point is that the lexicographical order is primarily defined for
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definition for countable number seems obvious, and the definition for an arbitrary ordinal seems like it should make sense.)
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Mathematics. As a professional
Mathematician of many years this brings me close to tears. I think Knowledge can do better.
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When the actual triples (caption in gray) are in lex order, the corresponding binary vectors (red caption) are in
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at OEIS-Wiki states that reverse lex. order is lex. order upside down. Reading the OEIS article I would say that
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The term "dictionary order" is a misnomer: most dictionaries are not in simple lexicographical order. --
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On the left side the smaller numbers are have a dark background, on the right side the bigger numbers.
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This is essentially said in section "Cartesian products". However, I agree that this section is
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on 2014-03-27. For the contribution history and old versions of the redirected page, please see
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That would be nice. Another observation is that these are all equivalent for any set with the
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though with that interpretation i would be appearing without any definition whatsoever.
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I have started cleaning up this article. I have a problem with the recent additions by
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b_{i}{\text{ for the largest }}i{\text{ for which }}a_{i}\neq b_{i}.}" /: -->
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The source of these additions are a wiki. Thus this is not a reliable source.
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b_{i}{\text{ for the largest }}i{\text{ for which }}a_{i}\neq b_{i}.}": -->
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Length-plus-lexicographic ordering: b < ab < aab < aaab < ...
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I'm removing the clarify tag, because it seems it's now been clarified.
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b_{i}{\text{ for the largest }}i{\text{ for which }}a_{i}\neq b_{i}.}
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colex order the corresponding inversion vectors are in colex order.
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Lexicographic ordering in multi-objective optimization
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I corrected a parsing error, and this is the result:
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1390:Parsing error
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1362:216.36.134.73
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1260:too technical
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880:At the moment
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871:Compare with
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1393:
1356:— Preceding
1352:
1335:67.198.37.16
1317:
1290:— Preceding
1283:
1212:89.93.179.93
1206:— Preceding
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268:Mid-priority
267:
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193:Mid‑priority
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102:
40:WikiProjects
988:Right side:
818:—Preceding
607:TooMuchMath
315:its history
243:Mathematics
234:mathematics
190:Mathematics
118:Linguistics
109:linguistics
59:Linguistics
30:Start-class
1564:Categories
1505:for which
1157:There are
1028:Victor Yus
986:lex order.
980:Left side:
967:Victor Yus
936:Victor Yus
641:questions?
585:aaab : -->
543:<< w
449:aaab : -->
304:page were
1130:sequences
1111:MGMontini
1103:MGMontini
1044:Tartolamp
884:Orderings
873:Orderings
654:Sipser's
584:aab : -->
448:aab : -->
335:The Anome
1358:unsigned
1304:contribs
1292:unsigned
1264:D.Lazard
1208:unsigned
1186:D.Lazard
1137:D.Lazard
1000:Compare:
858:Melchoir
820:unsigned
692:NavarroJ
583:ab : -->
447:ab : -->
372:Revolver
359:Revolver
344:Revolver
329:Untitled
1376:Elphion
1320:Elphion
1229:Elphion
1152:Frobitz
1082:Elphion
1012:Lipedia
952:Lipedia
892:Lipedia
796:Patrick
723:Please
621:Calbaer
270:on the
145:on the
1296:Qx2020
762:Beland
531:i<j
524:< w
306:merged
36:scale.
1548:TMM53
1483:: -->
518:then
308:into
1552:talk
1467:and
1380:talk
1366:talk
1339:talk
1324:talk
1300:talk
1268:talk
1250:talk
1233:talk
1216:talk
1190:talk
1171:and
1141:talk
1115:talk
1107:talk
1086:talk
1071:talk
1048:talk
1032:talk
1016:talk
971:talk
956:talk
940:talk
924:talk
896:talk
862:talk
828:talk
800:talk
782:talk
669:talk
636:fudo
537:and
529:iff
513:...y
499:and
494:...x
1007:rev
984:rev
727:to
586:...
535:i=j
533:or
505:= y
486:= x
262:Mid
137:Mid
1566::
1554:)
1546:.
1519:≠
1448:⋯
1416:⋯
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1302:•
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619:.
557:Σ
466:∗
462:Σ
432:∗
428:Σ
389:Σ
62::
1550:(
1532:.
1527:i
1523:b
1514:i
1510:a
1501:i
1491:i
1487:b
1478:i
1474:a
1459:n
1455:b
1451:+
1445:+
1440:1
1436:b
1432:=
1427:n
1423:a
1419:+
1413:+
1408:1
1404:a
1378:(
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515:j
511:2
509:y
507:1
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501:w
496:i
492:2
490:x
488:1
484:1
482:w
407:}
404:b
401:,
398:a
395:{
392:=
322:.
274:.
169:.
149:.
42::
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