2969:
1951:
4430:
25:
1457:
535:, a slightly different variant of the lexicographical order is used: the parts on the left of the decimal point are compared as before; if they are equal, the parts at the right of the decimal point are compared with the lexicographical order. The padding 'blank' in this context is a trailing "0" digit.
3843:
3847:
For this ordering, the monomials of degree one have the same order as the corresponding indeterminates (this would not be the case if the reverse lexicographical order would be used). For comparing monomials in two variables of the same total degree, this order is the same as the lexicographic
459:
In lexicographical order, the word "Thomas" appears before "Thompson" because they first differ at the fifth letter ('a' and 'p'), and letter 'a' comes before the letter 'p' in the alphabet. Because it is the first difference, in this case the 5th letter is the "most significant difference" for
3328:
if the monoid operation is denoted multiplicatively. This compatibility implies that the product of a polynomial by a monomial does not change the order of the terms. For Gröbner bases, a further condition must be satisfied, namely that every non-constant monomial is greater than the monomial
1318:
3000:
is a variant of the lexicographical order that is obtained by reading finite sequences from the right to the left instead of reading them from the left to the right. More precisely, whereas the lexicographical order between two sequences is defined by
3701:
3187:
In general, the difference between the colexicographical order and the lexicographical order is not very significant. However, when considering increasing sequences, typically for coding subsets, the two orders differ significantly.
873:
3520:, and then resolving the conflicts by using the lexicographical order. This order is not widely used, as either the lexicographical order or the degree reverse lexicographical order have generally better properties.
2500:
179:
There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements.
524:, and a natural number is larger than another one if either it has more digits (ignoring leading zeroes) or the number of digits is the same and the first (most significant) digit which differs is larger.
3696:
234:, that depends on the underlying ordering of the alphabet of symbols used to build the words. The lexicographical order is one way of formalizing word order given the order of the underlying symbols.
557:
standard for dates, which expresses a date as YYYY-MM-DD. This formatting scheme has the advantage that the lexicographical order on sequences of characters that represent dates coincides with the
3527:
consists also in comparing first the total degrees, and, in case of equality of the total degrees, using the reverse of the colexicographical order. That is, given two exponent vectors, one has
3326:
2048:
numbered in colexicographical order, and starting from 0. The lexicographical (lex) and colexicographical (colex) orders are on the top and the corresponding reverse orders (rev) on the bottom
1235:
As evaluating the lexicographical order of sequences compares only elements which have the same rank in the sequences, the lexicographical order extends to
Cartesian products of ordered sets.
450:, another convention is frequently used for the second case, whereby a shorter sequence is always smaller than a longer sequence. This variant of the lexicographical order is sometimes called
1163:
3405:
3848:
order. This is not the case with more variables. For example, for exponent vectors of monomials of degree two in three variables, one has for the degree reverse lexicographic order:
4167:
4004:
3504:
2958:
2727:
2692:
2656:
2568:
2534:
2210:
1602:
3469:
3440:
2906:
2851:
2822:
2624:
2439:
2355:
2046:
1997:
1452:{\displaystyle (a,b)\leq \left(a^{\prime },b^{\prime }\right){\text{ if and only if }}a<a^{\prime }{\text{ or }}\left(a=a^{\prime }{\text{ and }}b\leq b^{\prime }\right),}
1878:
681:
2132:
is equivalent to convert it into an increasing sequence. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the
893:
653:
436:
If two words have different lengths, the usual lexicographical order pads the shorter one with "blanks" (a special symbol that is treated as smaller than every element of
329:
2293:
1313:
2926:
2874:
2793:
1560:
918:
2753:
2319:
1655:
1095:
4206:
1210:
538:
When negative numbers are also considered, one has to reverse the order for comparing negative numbers. This is not usually a problem for humans, but it may be for
3615:
3509:
One of these admissible orders is the lexicographical order. It is, historically, the first to have been used for defining Gröbner bases, and is sometimes called
2382:
2106:
2083:
1811:
1768:
1282:
1233:
964:
941:
2773:
2588:
2402:
2264:
2130:
1940:
1920:
1900:
1831:
1788:
1745:
1725:
1702:
1622:
1499:
1479:
1259:
1183:
988:
3838:{\displaystyle a_{1}+\cdots +a_{n}=b_{1}+\cdots +b_{n}\quad {\text{ and }}\quad a_{i}>b_{i}{\text{ for the largest }}i{\text{ for which }}a_{i}\neq b_{i}.}
2448:
4216:
1523:
Unlike the finite case, an infinite product of well-orders is not necessarily well-ordered by the lexicographical order. For instance, the set of
3620:
1516:
One can define similarly the lexicographic order on the
Cartesian product of an infinite family of ordered sets, if the family is indexed by the
833:
4175:
is a multiple of the least indeterminate if and only if its leading monomial (its greater monomial) is a multiple of this least indeterminate.
4307:
4279:
2662:(it may be also injective if the forms are dependent, see below). The lexicographic order on the image of this map induces a group order on
42:
1520:, or more generally by a well-ordered set. This generalized lexicographical order is a total order if each factor set is totally ordered.
489:
finite words is well-ordered; for example, the infinite set of words {b, ab, aab, aaab, ... } has no lexicographically earliest element.
311:, including words of length 1 containing a single symbol, words of length 2 with 2 symbols, and so on, even including the empty sequence
4434:
2050:
One passes from an order to its reverse order, either by reading bottom-up instead of up-bottom, or by exchanging red and white colors.
1049:
Since many applications require well orders, a variant of the lexicographical orders is often used. This well-order, sometimes called
4351:
108:
89:
2977:
509:
61:
2266:
natural numbers. This is not the case for the lexicographical order, as, with the lexicographical order, we have, for example,
3191:
For example, for ordering the increasing sequences (or the sets) of two natural integers, the lexicographical order begins by
1119:
of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a
Cartesian product
3340:
As Gröbner bases are defined for polynomials in a fixed number of variables, it is common to identify monomials (for example
46:
4011:
68:
3851:
561:: an earlier CE date is smaller in the lexicographical order than a later date up to year 9999. This date ordering makes
3333:. However this condition is not needed for other related algorithms, such as the algorithms for the computation of the
3282:
2150:
For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of
479:
by the lexicographical order (provided the alphabet is finite); that is, every decreasing sequence of words of length
247:
3226:
1122:
75:
331:
with no symbols at all. The lexicographical order on the set of all these finite words orders the words as follows:
4200:
3343:
35:
4194:
1662:
610:
130:
4189:
520:
of the lexicographic order. In fact, with positional notation, a natural number is represented by a sequence of
57:
3252:
3196:
12 < 13 < 14 < 15 < ... < 23 < 24 < 25 < ... < 34 < 35 < ... < 45 < ...
504:
is that it is not always immediately obvious which of two numbers is the smaller. On the other hand, with the
3217:
is finite. In other words, the colexicographical order for increasing sequences of a given length induces an
4211:
3256:
2973:
1505:, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a
4172:
3530:
3213:
The main property of the colexicographical order for increasing sequences of a given length is that every
442:) at the end until the words are the same length, and then the words are compared as in the previous case.
4455:
4450:
2409:
1239:
485:
is finite (or equivalently, every non-empty subset has a least element). It is not true that the set of
211:
3477:
2931:
2700:
2665:
2629:
2541:
2507:
2153:
1568:
686:
With this terminology, the above definition of the lexicographical order becomes more concise: Given a
214:; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
3445:
3416:
2882:
2827:
2798:
2600:
2415:
2331:
2010:
1961:
4226:
2659:
501:
497:
The lexicographical order is used not only in dictionaries, but also commonly for numbers and dates.
1836:
3247:, the order of the terms does not matter in general, as the addition is commutative. However, some
2968:
1705:
1666:
663:
558:
543:
505:
173:
16:
Generalization of the alphabetical order of dictionaries to sequences of elements of an ordered set
4343:
878:
638:
314:
4407:
4323:
2358:
2000:
1524:
161:
126:
122:
82:
2269:
1292:
391:, the order of the two words depends on the alphabetic order of the symbols in the first place
4347:
4303:
4275:
3218:
2911:
2859:
2778:
2243:
1530:
1116:
901:
562:
207:
3221:
with the natural numbers, and allows enumerating these sequences. This is frequently used in
2732:
2298:
1631:
4397:
4357:
4327:
3260:
2004:
1682:
1506:
1074:
532:
1188:
4361:
4339:
3513:
for distinguishing it from other orders that are also related to a lexicographical order.
3214:
2239:
2147:. Therefore, in the following, we will consider only orders on subsets of fixed cardinal.
1006:
521:
251:
4328:
2217:
123 < 124 < 125 < 126 < 134 < 135 < 136 < 145 < 146 < 156 <
2364:
2088:
2065:
1793:
1750:
1264:
1215:
946:
923:
4242:
3264:
3238:
2758:
2573:
2405:
2387:
2249:
2231:
2144:
2115:
1925:
1905:
1885:
1816:
1773:
1730:
1710:
1687:
1607:
1517:
1484:
1464:
1244:
1168:
973:
734:
under lexicographical order, if at least one of the following conditions is satisfied:
599:(including the empty sequence, of length 0), and the operation (multiplication) is the
547:
517:
513:
452:
4429:
1950:
593:. That is, the elements of the monoid are the finite sequences (words) of elements of
4444:
4221:
3222:
2928:
is an order isomorphism when the image is equipped with the lexicographical order on
2059:
2055:
1510:
687:
600:
447:
231:
184:
4411:
2222:
234 < 235 < 236 < 245 < 246 < 256 < 345 < 346 < 356 < 456
4236:
3517:
3334:
3207:
12 < 13 < 23 < 14 < 24 < 34 < 15 < 25 < 35 < 45 < ...
1563:
1097:), and, if the lengths are equal, using the lexicographical order. If the order on
553:
Another example of a non-dictionary use of lexicographical ordering appears in the
227:
165:
2538:
The lexicographical ordering may also be used to characterize all group orders on
1770:
They can thus be ordered by the lexicographical order, and for two such functions
4247:
4231:
3268:
2594:
2590:
2442:
2140:
1999:
represented as sets of red squares, increasing sequences (in blue), or by their
1502:
691:
584:
528:
141:
24:
3255:, require the terms to be in a specific order. Many of the main algorithms for
2007:(in grey). The grey numbers are also the rank of the subsets in all subsets of
226:(the set of words used in some language) have a conventional ordering, used in
3244:
1813:
the lexicographical order is thus determined by their values for the smallest
1527:
binary sequences (by definition, the set of functions from natural numbers to
967:
476:
238:
192:
4184:
3413:
is the number of variables, every monomial order is thus the restriction to
3276:
3248:
542:(testing the sign takes some time). This is one of the reasons for adopting
134:
1628:) does not have a least element under the lexicographical order induced by
2694:
Robbiano's theorem is that every group order may be obtained in this way.
868:{\displaystyle \varepsilon <b\,\,{\text{ for all }}b\neq \varepsilon ,}
195:
by assigning a total order to the finite set, and converting subsets into
1051:
554:
539:
196:
169:
4402:
4203:- an algorithmic problem of finding a lexicographically-maximal element.
4171:
A useful property of the degree reverse lexicographical order is that a
4008:
For the lexicographical order, the same exponent vectors are ordered as
1922:
is finite, then the resulting order is a well-order. As shown above, if
655:) is a prefix of every word, and every word is a prefix of itself (with
397:
where the two words differ (counting from the beginning of the words):
4388:
Weispfenning, Volker (May 1987), "Admissible Orders and Linear Forms",
2109:
1604:) is not well-ordered; the subset of sequences that have precisely one
565:
of dates easier by avoiding the need for a separate sorting algorithm.
223:
2495:{\displaystyle a<b\quad {\text{ if and only if }}\quad a+c<b+c.}
3272:
1955:
188:
463:
An important property of the lexicographical order is that for each
2139:
In this context, one generally prefer to sort first the subsets by
4335:
4374:
Robbiano, L. (1985). Term orderings on the polynomial ring. In
2775:
linear forms with real coefficients, such that the induced map
121:
For similarly named ordering systems outside mathematics, see
18:
1059:, consists in considering first the lengths of the words (if
830:
Notice that, due to the prefix condition in this definition,
2242:
are finite, and thus the colexicographical order defines an
1436:
1415:
1389:
1363:
1350:
3691:{\displaystyle a_{1}+\cdots +a_{n}<b_{1}+\cdots +b_{n},}
2230:
For ordering finite subsets of a given cardinality of the
1103:
is a well-order, the same is true for the shortlex order.
512:, comparing numbers is easy, because the natural order on
2058:, one has often to enumerate, and therefore to order the
683:); care must be taken if these cases are to be excluded.
4212:
Lexicographic ordering in tensor abstract index notation
2238:
order (see below) is often more convenient, because all
172:
of ordered symbols or, more generally, of elements of a
4248:
Orders on the
Cartesian product of totally ordered sets
2972:
Orderings of the 24 permutations of {1,...,5} that are
1665:. Similarly, the infinite lexicographic product is not
4330:
Information and randomness. An algorithmic perspective
4197:- an application of lexicographic order in economics.
4014:
3854:
3704:
3623:
3533:
3480:
3448:
3419:
3346:
3285:
2934:
2914:
2885:
2862:
2830:
2801:
2781:
2761:
2735:
2703:
2668:
2632:
2603:
2576:
2544:
2510:
2451:
2418:
2390:
2367:
2334:
2301:
2272:
2252:
2156:
2118:
2091:
2068:
2013:
1964:
1928:
1908:
1888:
1839:
1819:
1796:
1776:
1753:
1733:
1713:
1690:
1634:
1610:
1571:
1533:
1487:
1467:
1321:
1295:
1267:
1247:
1218:
1191:
1171:
1125:
1077:
976:
949:
926:
904:
881:
836:
666:
641:
317:
2246:between the natural numbers and the set of sets of
1031:has no least element in the lexicographical order:
943:then so is the lexicographic order on the words of
49:. Unsourced material may be challenged and removed.
4293:
4291:
4161:
3998:
3837:
3690:
3609:
3498:
3463:
3434:
3399:
3320:
2952:
2920:
2900:
2868:
2845:
2816:
2787:
2767:
2747:
2721:
2686:
2650:
2618:
2582:
2562:
2528:
2494:
2433:
2396:
2376:
2349:
2313:
2287:
2258:
2204:
2124:
2100:
2077:
2040:
1991:
1934:
1914:
1894:
1872:
1825:
1805:
1782:
1762:
1739:
1719:
1696:
1649:
1616:
1596:
1554:
1493:
1473:
1451:
1307:
1276:
1253:
1227:
1204:
1177:
1157:
1089:
982:
958:
935:
912:
887:
867:
675:
647:
335:Given two different words of the same length, say
323:
199:, to which the lexicographical order is applied.
3321:{\displaystyle a<b{\text{ implies }}ac<bc,}
2504:The lexicographical ordering is a group order on
1671:011111... < 101111... < 110111 ... < ...
1111:The lexicographical order defines an order on an
1659:100000... > 010000... > 001000... > ...
4207:Lexicographic order topology on the unit square
1158:{\displaystyle E_{1}\times \cdots \times E_{n}}
4302:. Cambridge University Press. pp. 18–19.
2445:, which is compatible with addition, that is
1287:lexicographical order on the Cartesian product
4239:, a different way of combining partial orders
8:
4338:Monographs on Theoretical Computer Science.
3516:Another one consists in comparing first the
3400:{\displaystyle x_{1}x_{2}^{3}x_{4}x_{5}^{2}}
2199:
2163:
2032:
2014:
1983:
1965:
1727:may be identified with sequences indexed by
1585:
1572:
1546:
1534:
4378:(pp. 513-517). Springer Berlin Heidelberg.
4265:
4263:
3407:) with their exponent vectors (here ). If
3094:the colexicographical order is defined by
2085:For this, one usually chooses an order on
4401:
4217:Lexicographically minimal string rotation
4013:
3853:
3826:
3813:
3804:
3796:
3790:
3777:
3767:
3760:
3741:
3728:
3709:
3703:
3679:
3660:
3647:
3628:
3622:
3598:
3579:
3560:
3541:
3532:
3487:
3483:
3482:
3479:
3455:
3451:
3450:
3447:
3426:
3422:
3421:
3418:
3391:
3386:
3376:
3366:
3361:
3351:
3345:
3295:
3284:
2941:
2937:
2936:
2933:
2913:
2892:
2888:
2887:
2884:
2861:
2837:
2833:
2832:
2829:
2808:
2804:
2803:
2800:
2780:
2760:
2734:
2710:
2706:
2705:
2702:
2675:
2671:
2670:
2667:
2639:
2635:
2634:
2631:
2610:
2606:
2605:
2602:
2575:
2551:
2547:
2546:
2543:
2517:
2513:
2512:
2509:
2462:
2450:
2425:
2421:
2420:
2417:
2389:
2366:
2341:
2337:
2336:
2333:
2300:
2271:
2251:
2155:
2117:
2090:
2067:
2012:
1963:
1927:
1907:
1887:
1838:
1818:
1795:
1775:
1752:
1732:
1712:
1689:
1633:
1609:
1588:
1570:
1532:
1486:
1466:
1435:
1420:
1414:
1394:
1388:
1373:
1362:
1349:
1320:
1294:
1266:
1246:
1217:
1196:
1190:
1170:
1149:
1130:
1124:
1076:
975:
948:
925:
909:
905:
903:
880:
848:
847:
846:
835:
665:
640:
316:
305:are the finite sequences of symbols from
109:Learn how and when to remove this message
3263:, concept that requires the choice of a
3202:and the colexicographic order begins by
2967:
1949:
1626:{ 100000..., 010000..., 001000..., ... }
427:in the underlying order of the alphabet
202:A generalization defines an order on an
4435:Lexicographic and colexicographic order
4376:European Conference on Computer Algebra
4259:
2697:More precisely, given a group order on
635:. By this definition, the empty word (
4162:{\displaystyle <<<<<.}
272:that are not the same symbol, either
7:
4396:(2), New York, NY, USA: ACM: 16–18,
4298:Franz Baader; Tobias Nipkow (1999).
3999:{\displaystyle <<<<<}
3525:degree reverse lexicographical order
3472:
2658:which is injective if the forms are
47:adding citations to reliable sources
3225:, for example in the proof of the
1942:is infinite this is not the case.
1461:The result is a partial order. If
990:is well-ordered. For instance, if
966:However, in general this is not a
613:(or 'truncation') of another word
14:
3499:{\displaystyle \mathbb {Z} ^{n},}
2953:{\displaystyle \mathbb {R} ^{s}.}
2722:{\displaystyle \mathbb {Z} ^{n},}
2687:{\displaystyle \mathbb {Z} ^{n}.}
2651:{\displaystyle \mathbb {R} ^{n},}
2563:{\displaystyle \mathbb {Z} ^{n}.}
2529:{\displaystyle \mathbb {Z} ^{n}.}
2205:{\displaystyle S=\{1,2,3,4,5,6\}}
1677:Functions over a well-ordered set
1597:{\displaystyle \{0,1\}^{\omega }}
4428:
3464:{\displaystyle \mathbb {Z} ^{n}}
3435:{\displaystyle \mathbb {N} ^{n}}
2901:{\displaystyle \mathbb {Z} ^{n}}
2846:{\displaystyle \mathbb {R} ^{s}}
2817:{\displaystyle \mathbb {Z} ^{n}}
2619:{\displaystyle \mathbb {Z} ^{n}}
2597:coefficients, define a map from
2434:{\displaystyle \mathbb {Z} ^{n}}
2384:whose elements are sequences of
2350:{\displaystyle \mathbb {Z} ^{n}}
2041:{\displaystyle \{1,\ldots ,6\},}
1992:{\displaystyle \{1,\ldots ,6\},}
1673:is an infinite ascending chain.
546:representation for representing
237:The formal notion starts with a
183:Another variant, widely used in
23:
3772:
3766:
3279:. Here "compatible" means that
3271:, which is compatible with the
2879:the resulting isomorphism from
2467:
2461:
2404:integers, and operation is the
254:. That is, for any two symbols
34:needs additional citations for
4433:Learning materials related to
4153:
4135:
4129:
4111:
4105:
4087:
4081:
4063:
4057:
4039:
4033:
4015:
3993:
3975:
3969:
3951:
3945:
3927:
3921:
3903:
3897:
3879:
3873:
3855:
3604:
3572:
3566:
3534:
2853:has the following properties;
1873:{\displaystyle f(x)\neq g(x).}
1864:
1858:
1849:
1843:
1334:
1322:
769:(possibly empty) and elements
1:
676:{\displaystyle =\varepsilon }
469:, the set of words of length
160:) is a generalization of the
4274:. Springer. pp. 88–89.
2980:(in red) of permutations in
888:{\displaystyle \varepsilon }
648:{\displaystyle \varepsilon }
500:One of the drawbacks of the
324:{\displaystyle \varepsilon }
4300:Term Rewriting and All That
3798: for the largest
1057:quasi-lexicographical order
724:is non-empty, then one has
516:is the same as the variant
510:Hindu–Arabic numeral system
4472:
4201:Lexicographic optimization
3511:pure lexicographical order
3236:
2464: if and only if
2288:{\displaystyle 12n<134}
1375: if and only if
120:
4195:Lexicographic preferences
1902:is also well-ordered and
1663:infinite descending chain
1308:{\displaystyle A\times B}
493:Numeral systems and dates
131:lexicographic preferences
4270:Egbert Harzheim (2006).
3257:multivariate polynomials
3253:polynomial long division
2921:{\displaystyle \varphi }
2869:{\displaystyle \varphi }
2788:{\displaystyle \varphi }
1555:{\displaystyle \{0,1\},}
1238:Specifically, given two
913:{\displaystyle \,<\,}
3506:for a classification).
3442:of a monomial order of
2748:{\displaystyle s\leq n}
2729:there exist an integer
2314:{\displaystyle n>2.}
1650:{\displaystyle 0<1,}
970:, even if the alphabet
619:if there exists a word
460:alphabetical ordering.
4173:homogeneous polynomial
4163:
4000:
3839:
3692:
3611:
3500:
3473:§ Group orders of
3465:
3436:
3401:
3322:
3227:Kruskal–Katona theorem
2989:
2988:order, and vice versa.
2954:
2922:
2902:
2870:
2847:
2818:
2789:
2769:
2749:
2723:
2688:
2652:
2620:
2584:
2564:
2530:
2496:
2435:
2398:
2378:
2351:
2315:
2289:
2260:
2206:
2126:
2102:
2079:
2051:
2042:
1993:
1936:
1916:
1896:
1874:
1827:
1807:
1784:
1764:
1741:
1721:
1698:
1651:
1618:
1598:
1556:
1495:
1475:
1453:
1309:
1278:
1255:
1240:partially ordered sets
1229:
1206:
1179:
1159:
1091:
1090:{\displaystyle a<b}
984:
960:
937:
914:
889:
869:
677:
649:
325:
212:partially ordered sets
4164:
4001:
3840:
3806: for which
3693:
3612:
3501:
3466:
3437:
3402:
3323:
2971:
2964:Colexicographic order
2955:
2923:
2903:
2871:
2848:
2819:
2790:
2770:
2750:
2724:
2689:
2653:
2621:
2585:
2565:
2531:
2497:
2436:
2399:
2379:
2352:
2316:
2290:
2261:
2207:
2134:lexicographical order
2127:
2103:
2080:
2043:
1994:
1953:
1937:
1917:
1897:
1875:
1828:
1808:
1785:
1765:
1742:
1722:
1699:
1681:The functions from a
1652:
1619:
1599:
1557:
1496:
1476:
1454:
1310:
1279:
1256:
1230:
1207:
1205:{\displaystyle E_{i}}
1180:
1160:
1092:
985:
961:
938:
915:
890:
870:
678:
650:
326:
150:lexicographical order
58:"Lexicographic order"
4227:Long line (topology)
4190:Kleene–Brouwer order
4012:
3852:
3702:
3621:
3610:{\displaystyle <}
3531:
3478:
3446:
3417:
3344:
3283:
2932:
2912:
2883:
2860:
2828:
2799:
2779:
2759:
2733:
2701:
2666:
2660:linearly independent
2630:
2601:
2574:
2542:
2508:
2449:
2416:
2388:
2365:
2332:
2299:
2270:
2250:
2154:
2116:
2089:
2066:
2011:
1962:
1926:
1906:
1886:
1837:
1817:
1794:
1774:
1751:
1731:
1711:
1688:
1632:
1608:
1569:
1531:
1485:
1465:
1319:
1293:
1265:
1245:
1216:
1189:
1169:
1165:is a sequence whose
1123:
1075:
974:
947:
924:
920:is a total order on
902:
879:
834:
664:
639:
563:computerized sorting
502:Roman numeral system
315:
197:increasing sequences
43:improve this article
4403:10.1145/24554.24557
3396:
3371:
3297: implies
2001:indicator functions
1954:Orderings of the 3-
1706:totally ordered set
1185:element belongs to
895:is the empty word.
850: for all
751:there exists words
559:chronological order
506:positional notation
246:, often called the
174:totally ordered set
4159:
3996:
3835:
3688:
3607:
3496:
3461:
3432:
3397:
3382:
3357:
3318:
2990:
2950:
2918:
2898:
2866:
2843:
2814:
2785:
2765:
2745:
2719:
2684:
2648:
2616:
2580:
2560:
2526:
2492:
2431:
2394:
2377:{\displaystyle n,}
2374:
2359:free Abelian group
2347:
2311:
2285:
2256:
2202:
2122:
2101:{\displaystyle S.}
2098:
2078:{\displaystyle S.}
2075:
2052:
2038:
1989:
1932:
1912:
1892:
1870:
1823:
1806:{\displaystyle g,}
1803:
1780:
1763:{\displaystyle Y.}
1760:
1737:
1717:
1694:
1647:
1614:
1594:
1562:also known as the
1552:
1525:countably infinite
1491:
1471:
1449:
1305:
1277:{\displaystyle B,}
1274:
1251:
1228:{\displaystyle i.}
1225:
1202:
1175:
1155:
1107:Cartesian products
1087:
980:
959:{\displaystyle A.}
956:
936:{\displaystyle A,}
933:
910:
885:
865:
673:
645:
321:
162:alphabetical order
127:natural sort order
123:alphabetical order
4309:978-0-521-77920-3
4281:978-0-387-24222-4
3807:
3799:
3770:
3298:
3275:structure of the
3259:are related with
3243:When considering
3219:order isomorphism
2978:inversion vectors
2768:{\displaystyle s}
2583:{\displaystyle n}
2465:
2397:{\displaystyle n}
2324:Group orders of Z
2259:{\displaystyle n}
2244:order isomorphism
2236:colexicographical
2143:, such as in the
2125:{\displaystyle S}
1935:{\displaystyle X}
1915:{\displaystyle X}
1895:{\displaystyle Y}
1826:{\displaystyle x}
1783:{\displaystyle f}
1740:{\displaystyle X}
1720:{\displaystyle Y}
1697:{\displaystyle X}
1617:{\displaystyle 1}
1494:{\displaystyle B}
1474:{\displaystyle A}
1423:
1397:
1376:
1254:{\displaystyle A}
1178:{\displaystyle i}
1117:Cartesian product
983:{\displaystyle A}
851:
603:of words. A word
577:over an alphabet
208:Cartesian product
119:
118:
111:
93:
4463:
4432:
4416:
4414:
4405:
4385:
4379:
4372:
4366:
4365:
4333:
4324:Calude, Cristian
4320:
4314:
4313:
4295:
4286:
4285:
4267:
4168:
4166:
4165:
4160:
4005:
4003:
4002:
3997:
3844:
3842:
3841:
3836:
3831:
3830:
3818:
3817:
3808:
3805:
3800:
3797:
3795:
3794:
3782:
3781:
3771:
3768:
3765:
3764:
3746:
3745:
3733:
3732:
3714:
3713:
3697:
3695:
3694:
3689:
3684:
3683:
3665:
3664:
3652:
3651:
3633:
3632:
3616:
3614:
3613:
3608:
3603:
3602:
3584:
3583:
3565:
3564:
3546:
3545:
3505:
3503:
3502:
3497:
3492:
3491:
3486:
3470:
3468:
3467:
3462:
3460:
3459:
3454:
3441:
3439:
3438:
3433:
3431:
3430:
3425:
3412:
3406:
3404:
3403:
3398:
3395:
3390:
3381:
3380:
3370:
3365:
3356:
3355:
3332:
3327:
3325:
3324:
3319:
3299:
3296:
3208:
3197:
3182:
3173:
3164:
3158:
3142:
3089:
3080:
3071:
3065:
3049:
2959:
2957:
2956:
2951:
2946:
2945:
2940:
2927:
2925:
2924:
2919:
2908:to the image of
2907:
2905:
2904:
2899:
2897:
2896:
2891:
2875:
2873:
2872:
2867:
2852:
2850:
2849:
2844:
2842:
2841:
2836:
2823:
2821:
2820:
2815:
2813:
2812:
2807:
2794:
2792:
2791:
2786:
2774:
2772:
2771:
2766:
2754:
2752:
2751:
2746:
2728:
2726:
2725:
2720:
2715:
2714:
2709:
2693:
2691:
2690:
2685:
2680:
2679:
2674:
2657:
2655:
2654:
2649:
2644:
2643:
2638:
2625:
2623:
2622:
2617:
2615:
2614:
2609:
2589:
2587:
2586:
2581:
2569:
2567:
2566:
2561:
2556:
2555:
2550:
2535:
2533:
2532:
2527:
2522:
2521:
2516:
2501:
2499:
2498:
2493:
2466:
2463:
2440:
2438:
2437:
2432:
2430:
2429:
2424:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2356:
2354:
2353:
2348:
2346:
2345:
2340:
2320:
2318:
2317:
2312:
2294:
2292:
2291:
2286:
2265:
2263:
2262:
2257:
2240:initial segments
2223:
2218:
2211:
2209:
2208:
2203:
2131:
2129:
2128:
2123:
2107:
2105:
2104:
2099:
2084:
2082:
2081:
2076:
2047:
2045:
2044:
2039:
2005:decimal notation
1998:
1996:
1995:
1990:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1901:
1899:
1898:
1893:
1879:
1877:
1876:
1871:
1832:
1830:
1829:
1824:
1812:
1810:
1809:
1804:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1703:
1701:
1700:
1695:
1683:well-ordered set
1672:
1660:
1656:
1654:
1653:
1648:
1627:
1623:
1621:
1620:
1615:
1603:
1601:
1600:
1595:
1593:
1592:
1561:
1559:
1558:
1553:
1507:linear extension
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1458:
1456:
1455:
1450:
1445:
1441:
1440:
1439:
1424:
1421:
1419:
1418:
1398:
1395:
1393:
1392:
1377:
1374:
1372:
1368:
1367:
1366:
1354:
1353:
1314:
1312:
1311:
1306:
1289:
1288:
1283:
1281:
1280:
1275:
1260:
1258:
1257:
1252:
1234:
1232:
1231:
1226:
1211:
1209:
1208:
1203:
1201:
1200:
1184:
1182:
1181:
1176:
1164:
1162:
1161:
1156:
1154:
1153:
1135:
1134:
1102:
1096:
1094:
1093:
1088:
1070:
1045:
1030:
1004:
989:
987:
986:
981:
965:
963:
962:
957:
942:
940:
939:
934:
919:
917:
916:
911:
894:
892:
891:
886:
874:
872:
871:
866:
852:
849:
824:
813:
802:
786:
780:
774:
768:
762:
756:
748:
742:
733:
723:
717:
711:
705:
700:, and two words
699:
682:
680:
679:
674:
660:
654:
652:
651:
646:
634:
624:
618:
608:
598:
592:
582:
544:two's complement
533:decimal notation
522:numerical digits
484:
474:
468:
441:
432:
426:
406:
396:
390:
362:
330:
328:
327:
322:
310:
304:
291:
281:
271:
265:
259:
245:
158:dictionary order
114:
107:
103:
100:
94:
92:
51:
27:
19:
4471:
4470:
4466:
4465:
4464:
4462:
4461:
4460:
4441:
4440:
4425:
4420:
4419:
4390:SIGSAM Bulletin
4387:
4386:
4382:
4373:
4369:
4354:
4340:Springer-Verlag
4322:
4321:
4317:
4310:
4297:
4296:
4289:
4282:
4269:
4268:
4261:
4256:
4181:
4010:
4009:
3850:
3849:
3822:
3809:
3786:
3773:
3769: and
3756:
3737:
3724:
3705:
3700:
3699:
3675:
3656:
3643:
3624:
3619:
3618:
3594:
3575:
3556:
3537:
3529:
3528:
3481:
3476:
3475:
3449:
3444:
3443:
3420:
3415:
3414:
3408:
3372:
3347:
3342:
3341:
3330:
3281:
3280:
3241:
3235:
3215:initial segment
3206:
3195:
3180:
3175:
3171:
3166:
3160:
3156:
3149:
3144:
3141:
3132:
3126:
3119:
3110:
3104:
3098:
3087:
3082:
3078:
3073:
3067:
3063:
3056:
3051:
3048:
3039:
3033:
3026:
3017:
3011:
3005:
2994:colexicographic
2976:(in blue). The
2966:
2935:
2930:
2929:
2910:
2909:
2886:
2881:
2880:
2858:
2857:
2831:
2826:
2825:
2802:
2797:
2796:
2777:
2776:
2757:
2756:
2731:
2730:
2704:
2699:
2698:
2669:
2664:
2663:
2633:
2628:
2627:
2604:
2599:
2598:
2572:
2571:
2545:
2540:
2539:
2511:
2506:
2505:
2447:
2446:
2419:
2414:
2413:
2386:
2385:
2363:
2362:
2335:
2330:
2329:
2326:
2297:
2296:
2268:
2267:
2248:
2247:
2232:natural numbers
2221:
2216:
2152:
2151:
2114:
2113:
2087:
2086:
2064:
2063:
2062:of a given set
2049:
2009:
2008:
2003:, converted in
1960:
1959:
1948:
1924:
1923:
1904:
1903:
1884:
1883:
1835:
1834:
1815:
1814:
1792:
1791:
1772:
1771:
1749:
1748:
1747:of elements of
1729:
1728:
1709:
1708:
1686:
1685:
1679:
1670:
1669:either because
1658:
1630:
1629:
1625:
1606:
1605:
1584:
1567:
1566:
1529:
1528:
1518:natural numbers
1503:totally ordered
1483:
1482:
1463:
1462:
1431:
1422: and
1410:
1403:
1399:
1384:
1358:
1345:
1344:
1340:
1317:
1316:
1291:
1290:
1286:
1285:
1263:
1262:
1243:
1242:
1214:
1213:
1192:
1187:
1186:
1167:
1166:
1145:
1126:
1121:
1120:
1109:
1098:
1073:
1072:
1060:
1032:
1009:
991:
972:
971:
945:
944:
922:
921:
900:
899:
877:
876:
832:
831:
816:
805:
794:
782:
776:
770:
764:
758:
752:
744:
743:is a prefix of
738:
725:
719:
713:
707:
701:
695:
692:totally ordered
662:
661:
656:
637:
636:
626:
620:
614:
604:
594:
588:
578:
575:monoid of words
571:
569:Monoid of words
548:signed integers
514:natural numbers
495:
480:
470:
464:
437:
428:
425:
416:
408:
407:if and only if
398:
392:
389:
380:
374:
364:
361:
352:
346:
336:
313:
312:
306:
300:
283:
273:
267:
261:
255:
252:totally ordered
241:
222:The words in a
220:
152:(also known as
138:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4469:
4467:
4459:
4458:
4453:
4443:
4442:
4439:
4438:
4437:at Wikiversity
4424:
4423:External links
4421:
4418:
4417:
4380:
4367:
4352:
4315:
4308:
4287:
4280:
4258:
4257:
4255:
4252:
4251:
4250:
4245:
4243:Shortlex order
4240:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4198:
4192:
4187:
4180:
4177:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3834:
3829:
3825:
3821:
3816:
3812:
3803:
3793:
3789:
3785:
3780:
3776:
3763:
3759:
3755:
3752:
3749:
3744:
3740:
3736:
3731:
3727:
3723:
3720:
3717:
3712:
3708:
3687:
3682:
3678:
3674:
3671:
3668:
3663:
3659:
3655:
3650:
3646:
3642:
3639:
3636:
3631:
3627:
3606:
3601:
3597:
3593:
3590:
3587:
3582:
3578:
3574:
3571:
3568:
3563:
3559:
3555:
3552:
3549:
3544:
3540:
3536:
3526:
3512:
3495:
3490:
3485:
3458:
3453:
3429:
3424:
3394:
3389:
3385:
3379:
3375:
3369:
3364:
3360:
3354:
3350:
3317:
3314:
3311:
3308:
3305:
3302:
3294:
3291:
3288:
3265:monomial order
3239:Monomial order
3237:Main article:
3234:
3231:
3211:
3210:
3200:
3199:
3185:
3184:
3178:
3169:
3154:
3147:
3137:
3130:
3124:
3115:
3108:
3102:
3092:
3091:
3085:
3076:
3066:for the first
3061:
3054:
3044:
3037:
3031:
3022:
3015:
3009:
2965:
2962:
2961:
2960:
2949:
2944:
2939:
2917:
2895:
2890:
2877:
2865:
2840:
2835:
2811:
2806:
2784:
2764:
2744:
2741:
2738:
2718:
2713:
2708:
2683:
2678:
2673:
2647:
2642:
2637:
2613:
2608:
2579:
2559:
2554:
2549:
2525:
2520:
2515:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2460:
2457:
2454:
2428:
2423:
2393:
2373:
2370:
2344:
2339:
2325:
2322:
2310:
2307:
2304:
2284:
2281:
2278:
2275:
2255:
2237:
2228:
2227:
2226:
2225:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2145:shortlex order
2135:
2121:
2097:
2094:
2074:
2071:
2060:finite subsets
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1947:
1946:Finite subsets
1944:
1931:
1911:
1891:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1822:
1802:
1799:
1779:
1759:
1756:
1736:
1716:
1693:
1678:
1675:
1646:
1643:
1640:
1637:
1613:
1591:
1587:
1583:
1580:
1577:
1574:
1551:
1548:
1545:
1542:
1539:
1536:
1490:
1470:
1448:
1444:
1438:
1434:
1430:
1427:
1417:
1413:
1409:
1406:
1402:
1396: or
1391:
1387:
1383:
1380:
1371:
1365:
1361:
1357:
1352:
1348:
1343:
1339:
1336:
1333:
1330:
1327:
1324:
1315:is defined as
1304:
1301:
1298:
1273:
1270:
1250:
1224:
1221:
1199:
1195:
1174:
1152:
1148:
1144:
1141:
1138:
1133:
1129:
1108:
1105:
1086:
1083:
1080:
1065:) < length(
1058:
1054:
979:
955:
952:
932:
929:
908:
884:
864:
861:
858:
855:
845:
842:
839:
828:
827:
826:
825:
814:
803:
789:
788:
749:
672:
669:
644:
576:
570:
567:
550:in computers.
494:
491:
455:
453:shortlex order
444:
443:
434:
421:
412:
385:
378:
372:
357:
350:
344:
320:
219:
216:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4468:
4457:
4454:
4452:
4449:
4448:
4446:
4436:
4431:
4427:
4426:
4422:
4413:
4409:
4404:
4399:
4395:
4391:
4384:
4381:
4377:
4371:
4368:
4363:
4359:
4355:
4353:3-540-57456-5
4349:
4345:
4341:
4337:
4332:
4331:
4325:
4319:
4316:
4311:
4305:
4301:
4294:
4292:
4288:
4283:
4277:
4273:
4266:
4264:
4260:
4253:
4249:
4246:
4244:
4241:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4222:Leximin order
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4202:
4199:
4196:
4193:
4191:
4188:
4186:
4183:
4182:
4178:
4176:
4174:
4169:
4156:
4150:
4147:
4144:
4141:
4138:
4132:
4126:
4123:
4120:
4117:
4114:
4108:
4102:
4099:
4096:
4093:
4090:
4084:
4078:
4075:
4072:
4069:
4066:
4060:
4054:
4051:
4048:
4045:
4042:
4036:
4030:
4027:
4024:
4021:
4018:
4006:
3990:
3987:
3984:
3981:
3978:
3972:
3966:
3963:
3960:
3957:
3954:
3948:
3942:
3939:
3936:
3933:
3930:
3924:
3918:
3915:
3912:
3909:
3906:
3900:
3894:
3891:
3888:
3885:
3882:
3876:
3870:
3867:
3864:
3861:
3858:
3845:
3832:
3827:
3823:
3819:
3814:
3810:
3801:
3791:
3787:
3783:
3778:
3774:
3761:
3757:
3753:
3750:
3747:
3742:
3738:
3734:
3729:
3725:
3721:
3718:
3715:
3710:
3706:
3685:
3680:
3676:
3672:
3669:
3666:
3661:
3657:
3653:
3648:
3644:
3640:
3637:
3634:
3629:
3625:
3599:
3595:
3591:
3588:
3585:
3580:
3576:
3569:
3561:
3557:
3553:
3550:
3547:
3542:
3538:
3524:
3521:
3519:
3518:total degrees
3514:
3510:
3507:
3493:
3488:
3474:
3456:
3427:
3411:
3392:
3387:
3383:
3377:
3373:
3367:
3362:
3358:
3352:
3348:
3338:
3336:
3315:
3312:
3309:
3306:
3303:
3300:
3292:
3289:
3286:
3278:
3274:
3270:
3266:
3262:
3261:Gröbner bases
3258:
3254:
3250:
3246:
3240:
3232:
3230:
3228:
3224:
3223:combinatorics
3220:
3216:
3205:
3204:
3203:
3194:
3193:
3192:
3189:
3181:
3172:
3163:
3159:for the last
3157:
3150:
3140:
3136:
3129:
3123:
3118:
3114:
3107:
3101:
3097:
3096:
3095:
3088:
3079:
3070:
3064:
3057:
3047:
3043:
3036:
3030:
3025:
3021:
3014:
3008:
3004:
3003:
3002:
2999:
2995:
2987:
2984:order are in
2983:
2979:
2975:
2970:
2963:
2947:
2942:
2915:
2893:
2878:
2876:is injective;
2863:
2856:
2855:
2854:
2838:
2809:
2782:
2762:
2742:
2739:
2736:
2716:
2711:
2695:
2681:
2676:
2661:
2645:
2640:
2611:
2596:
2592:
2577:
2557:
2552:
2536:
2523:
2518:
2502:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2458:
2455:
2452:
2444:
2426:
2411:
2407:
2391:
2371:
2368:
2360:
2342:
2323:
2321:
2308:
2305:
2302:
2282:
2279:
2276:
2273:
2253:
2245:
2241:
2235:
2233:
2220:
2219:
2215:
2214:
2213:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2160:
2157:
2148:
2146:
2142:
2137:
2133:
2119:
2111:
2095:
2092:
2072:
2069:
2061:
2057:
2056:combinatorics
2035:
2029:
2026:
2023:
2020:
2017:
2006:
2002:
1986:
1980:
1977:
1974:
1971:
1968:
1957:
1952:
1945:
1943:
1929:
1909:
1889:
1880:
1867:
1861:
1855:
1852:
1846:
1840:
1820:
1800:
1797:
1777:
1757:
1754:
1734:
1714:
1707:
1691:
1684:
1676:
1674:
1668:
1664:
1644:
1641:
1638:
1635:
1611:
1589:
1581:
1578:
1575:
1565:
1549:
1543:
1540:
1537:
1526:
1521:
1519:
1514:
1512:
1511:product order
1508:
1504:
1488:
1468:
1459:
1446:
1442:
1432:
1428:
1425:
1411:
1407:
1404:
1400:
1385:
1381:
1378:
1369:
1359:
1355:
1346:
1341:
1337:
1331:
1328:
1325:
1302:
1299:
1296:
1271:
1268:
1248:
1241:
1236:
1222:
1219:
1197:
1193:
1172:
1150:
1146:
1142:
1139:
1136:
1131:
1127:
1118:
1114:
1106:
1104:
1101:
1084:
1081:
1078:
1068:
1064:
1056:
1053:
1050:
1047:
1044:
1040:
1036:
1028:
1024:
1020:
1016:
1013:
1008:
1002:
998:
994:
977:
969:
953:
950:
930:
927:
906:
896:
882:
862:
859:
856:
853:
843:
840:
837:
823:
819:
815:
812:
808:
804:
801:
797:
793:
792:
791:
790:
785:
779:
773:
767:
761:
755:
750:
747:
741:
737:
736:
735:
732:
728:
722:
716:
710:
704:
698:
693:
689:
684:
670:
667:
659:
642:
633:
629:
623:
617:
612:
607:
602:
601:concatenation
597:
591:
586:
581:
574:
568:
566:
564:
560:
556:
551:
549:
545:
541:
536:
534:
530:
525:
523:
519:
515:
511:
507:
503:
498:
492:
490:
488:
483:
478:
473:
467:
461:
457:
454:
451:
449:
448:combinatorics
440:
435:
431:
424:
420:
415:
411:
405:
401:
395:
388:
384:
377:
371:
367:
360:
356:
349:
343:
339:
334:
333:
332:
318:
309:
303:
298:
293:
290:
286:
280:
276:
270:
264:
258:
253:
249:
244:
240:
235:
233:
232:encyclopedias
229:
225:
217:
215:
213:
209:
205:
200:
198:
194:
190:
186:
185:combinatorics
181:
177:
175:
171:
167:
163:
159:
155:
154:lexical order
151:
147:
146:lexicographic
143:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4456:Lexicography
4451:Order theory
4393:
4389:
4383:
4375:
4370:
4329:
4318:
4299:
4272:Ordered Sets
4271:
4237:Star product
4170:
4007:
3846:
3522:
3515:
3508:
3409:
3339:
3335:tangent cone
3267:, that is a
3242:
3212:
3201:
3190:
3186:
3176:
3167:
3161:
3152:
3145:
3138:
3134:
3127:
3121:
3116:
3112:
3105:
3099:
3093:
3083:
3074:
3068:
3059:
3052:
3045:
3041:
3034:
3028:
3023:
3019:
3012:
3006:
2997:
2993:
2991:
2985:
2981:
2696:
2591:linear forms
2537:
2503:
2327:
2229:
2149:
2138:
2112:a subset of
2053:
1881:
1680:
1564:Cantor space
1522:
1515:
1460:
1237:
1112:
1110:
1099:
1066:
1062:
1048:
1042:
1038:
1034:
1026:
1022:
1018:
1014:
1011:
1000:
996:
992:
897:
829:
821:
817:
810:
806:
799:
795:
783:
777:
771:
765:
759:
753:
745:
739:
730:
726:
720:
714:
708:
702:
696:
685:
657:
631:
627:
621:
615:
605:
595:
589:
579:
572:
552:
537:
529:real numbers
526:
499:
496:
486:
481:
477:well-ordered
471:
465:
462:
458:
446:However, in
445:
438:
429:
422:
418:
413:
409:
403:
399:
393:
386:
382:
375:
369:
365:
358:
354:
347:
341:
337:
307:
301:
296:
294:
288:
284:
278:
274:
268:
262:
256:
242:
236:
228:dictionaries
221:
203:
201:
182:
178:
166:dictionaries
157:
153:
149:
145:
139:
105:
99:January 2022
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4232:Lyndon word
3617:if either
3471:(see above
3269:total order
3245:polynomials
2998:colex order
2443:total order
2410:group order
2141:cardinality
585:free monoid
531:written in
250:, which is
191:of a given
142:mathematics
4445:Categories
4362:0922.68073
4342:. p.
4254:References
3251:, such as
3249:algorithms
2295:for every
1833:such that
1667:Noetherian
1624:(that is,
1212:for every
968:well-order
718:such that
625:such that
239:finite set
218:Definition
193:finite set
69:newspapers
4185:Collation
3820:≠
3751:⋯
3719:⋯
3670:⋯
3638:⋯
3589:…
3551:…
3277:monomials
3233:Monomials
2916:φ
2864:φ
2783:φ
2740:≤
2570:In fact,
2024:…
1975:…
1853:≠
1590:ω
1509:of their
1501:are each
1437:′
1429:≤
1416:′
1390:′
1364:′
1351:′
1338:≤
1300:×
1143:×
1140:⋯
1137:×
1033:... <
883:ε
860:ε
857:≠
838:ε
787:such that
688:partially
671:ε
643:ε
540:computers
319:ε
187:, orders
170:sequences
135:collation
4412:10226875
4326:(1994).
4179:See also
2986:revcolex
2974:5-cycles
2406:addition
2361:of rank
1657:because
1052:shortlex
1007:language
555:ISO 8601
518:shortlex
248:alphabet
3090:differ,
2357:be the
2110:sorting
1956:subsets
1071:, then
1061:length(
583:is the
508:of the
224:lexicon
189:subsets
164:of the
83:scholar
4410:
4360:
4350:
4306:
4278:
3273:monoid
3183:differ
3165:where
3072:where
2234:, the
2108:Then,
1661:is an
1005:, the
875:where
611:prefix
144:, the
133:, and
85:
78:
71:
64:
56:
4408:S2CID
4336:EATCS
3151:<
3120:<
3058:<
3027:<
2982:colex
2824:into
2795:from
2626:into
2593:with
2441:is a
1704:to a
1115:-ary
1041:<
1037:<
1025:>
1021:≥ 0,
798:<
729:<
712:over
609:is a
587:over
417:<
402:<
297:words
287:<
277:<
206:-ary
156:, or
90:JSTOR
76:books
4348:ISBN
4304:ISBN
4276:ISBN
4133:<
4109:<
4085:<
4061:<
4037:<
3973:<
3949:<
3925:<
3901:<
3877:<
3784:>
3654:<
3570:<
3523:The
3307:<
3290:<
3174:and
3081:and
3040:...
2992:The
2755:and
2595:real
2478:<
2456:<
2408:. A
2328:Let
2306:>
2280:<
1790:and
1639:<
1481:and
1382:<
1284:the
1261:and
1082:<
907:<
841:<
775:and
706:and
694:set
573:The
527:For
363:and
295:The
260:and
230:and
62:news
4398:doi
4358:Zbl
3698:or
3143:if
3133:...
3111:...
3050:if
3018:...
2996:or
2412:on
2283:134
2212:is
2054:In
1958:of
1882:If
1055:or
1035:aab
995:= {
898:If
822:uyw
811:uxv
781:of
690:or
487:all
475:is
381:...
353:...
299:of
282:or
266:in
210:of
168:to
148:or
140:In
45:by
4447::
4406:,
4394:21
4392:,
4356:.
4346:.
4334:.
4290:^
4262:^
3337:.
3229:.
2309:2.
2274:12
2136:.
1513:.
1046:.
1039:ab
1017:|
999:,
820:=
809:=
763:,
757:,
632:uw
630:=
456:.
368:=
340:=
292:.
176:.
129:,
125:,
4415:.
4400::
4364:.
4344:1
4312:.
4284:.
4157:.
4154:]
4151:0
4148:,
4145:0
4142:,
4139:2
4136:[
4130:]
4127:0
4124:,
4121:1
4118:,
4115:1
4112:[
4106:]
4103:1
4100:,
4097:0
4094:,
4091:1
4088:[
4082:]
4079:0
4076:,
4073:2
4070:,
4067:0
4064:[
4058:]
4055:1
4052:,
4049:1
4046:,
4043:0
4040:[
4034:]
4031:2
4028:,
4025:0
4022:,
4019:0
4016:[
3994:]
3991:0
3988:,
3985:0
3982:,
3979:2
3976:[
3970:]
3967:0
3964:,
3961:1
3958:,
3955:1
3952:[
3946:]
3943:0
3940:,
3937:2
3934:,
3931:0
3928:[
3922:]
3919:1
3916:,
3913:0
3910:,
3907:1
3904:[
3898:]
3895:1
3892:,
3889:1
3886:,
3883:0
3880:[
3874:]
3871:2
3868:,
3865:0
3862:,
3859:0
3856:[
3833:.
3828:i
3824:b
3815:i
3811:a
3802:i
3792:i
3788:b
3779:i
3775:a
3762:n
3758:b
3754:+
3748:+
3743:1
3739:b
3735:=
3730:n
3726:a
3722:+
3716:+
3711:1
3707:a
3686:,
3681:n
3677:b
3673:+
3667:+
3662:1
3658:b
3649:n
3645:a
3641:+
3635:+
3630:1
3626:a
3605:]
3600:n
3596:b
3592:,
3586:,
3581:1
3577:b
3573:[
3567:]
3562:n
3558:a
3554:,
3548:,
3543:1
3539:a
3535:[
3494:,
3489:n
3484:Z
3457:n
3452:Z
3428:n
3423:N
3410:n
3393:2
3388:5
3384:x
3378:4
3374:x
3368:3
3363:2
3359:x
3353:1
3349:x
3331:1
3316:,
3313:c
3310:b
3304:c
3301:a
3293:b
3287:a
3209:.
3198:,
3179:i
3177:b
3170:i
3168:a
3162:i
3155:i
3153:b
3148:i
3146:a
3139:k
3135:b
3131:2
3128:b
3125:1
3122:b
3117:k
3113:a
3109:2
3106:a
3103:1
3100:a
3086:i
3084:b
3077:i
3075:a
3069:i
3062:i
3060:b
3055:i
3053:a
3046:k
3042:b
3038:2
3035:b
3032:1
3029:b
3024:k
3020:a
3016:2
3013:a
3010:1
3007:a
2948:.
2943:s
2938:R
2894:n
2889:Z
2839:s
2834:R
2810:n
2805:Z
2763:s
2743:n
2737:s
2717:,
2712:n
2707:Z
2682:.
2677:n
2672:Z
2646:,
2641:n
2636:R
2612:n
2607:Z
2578:n
2558:.
2553:n
2548:Z
2524:.
2519:n
2514:Z
2490:.
2487:c
2484:+
2481:b
2475:c
2472:+
2469:a
2459:b
2453:a
2427:n
2422:Z
2392:n
2372:,
2369:n
2343:n
2338:Z
2303:n
2277:n
2254:n
2224:.
2200:}
2197:6
2194:,
2191:5
2188:,
2185:4
2182:,
2179:3
2176:,
2173:2
2170:,
2167:1
2164:{
2161:=
2158:S
2120:S
2096:.
2093:S
2073:.
2070:S
2036:,
2033:}
2030:6
2027:,
2021:,
2018:1
2015:{
1987:,
1984:}
1981:6
1978:,
1972:,
1969:1
1966:{
1930:X
1910:X
1890:Y
1868:.
1865:)
1862:x
1859:(
1856:g
1850:)
1847:x
1844:(
1841:f
1821:x
1801:,
1798:g
1778:f
1758:.
1755:Y
1735:X
1715:Y
1692:X
1645:,
1642:1
1636:0
1612:1
1586:}
1582:1
1579:,
1576:0
1573:{
1550:,
1547:}
1544:1
1541:,
1538:0
1535:{
1489:B
1469:A
1447:,
1443:)
1433:b
1426:b
1412:a
1408:=
1405:a
1401:(
1386:a
1379:a
1370:)
1360:b
1356:,
1347:a
1342:(
1335:)
1332:b
1329:,
1326:a
1323:(
1303:B
1297:A
1272:,
1269:B
1249:A
1223:.
1220:i
1198:i
1194:E
1173:i
1151:n
1147:E
1132:1
1128:E
1113:n
1100:A
1085:b
1079:a
1069:)
1067:b
1063:a
1043:b
1029:}
1027:ε
1023:b
1019:n
1015:b
1012:a
1010:{
1003:}
1001:b
997:a
993:A
978:A
954:.
951:A
931:,
928:A
863:,
854:b
844:b
818:b
807:a
800:y
796:x
784:A
778:y
772:x
766:w
760:v
754:u
746:b
740:a
731:b
727:a
721:b
715:A
709:b
703:a
697:A
668:=
658:w
628:v
622:w
616:v
606:u
596:A
590:A
580:A
482:n
472:n
466:n
439:A
433:.
430:A
423:i
419:b
414:i
410:a
404:b
400:a
394:i
387:k
383:b
379:2
376:b
373:1
370:b
366:b
359:k
355:a
351:2
348:a
345:1
342:a
338:a
308:A
302:A
289:a
285:b
279:b
275:a
269:A
263:b
257:a
243:A
204:n
137:.
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
