2980:
1962:
4441:
36:
1468:
546:, 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.
3854:
3858:
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
470:
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
3339:
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
1329:
3011:
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
3712:
3198:
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.
884:
3531:, 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.
2511:
190:
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.
535:, 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.
3707:
245:, 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.
568:
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
3538:
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
3337:
2059:
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
1246:
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.
461:, 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
1174:
3416:
3859:
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:
4178:
4015:
3515:
2969:
2738:
2703:
2667:
2579:
2545:
2221:
1613:
3480:
3451:
2917:
2862:
2833:
2635:
2450:
2366:
2057:
2008:
1463:{\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),}
1889:
692:
2143:
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
904:
664:
447:
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
340:
2304:
1324:
2937:
2885:
2804:
1571:
929:
2764:
2330:
1666:
1106:
1221:
549:
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
3626:
3520:
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
2393:
2117:
2094:
1822:
1779:
1293:
1244:
975:
952:
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2599:
2413:
2275:
2141:
1951:
1931:
1911:
1842:
1799:
1756:
1736:
1713:
1633:
1510:
1490:
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1194:
999:
3849:{\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}.}
2459:
1534:
Unlike the finite case, an infinite product of well-orders is not necessarily well-ordered by the lexicographical order. For instance, the set of
4217:
3631:
1527:
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
844:
4186:
is a multiple of the least indeterminate if and only if its leading monomial (its greater monomial) is a multiple of this least indeterminate.
2673:(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
1531:, or more generally by a well-ordered set. This generalized lexicographical order is a total order if each factor set is totally ordered.
500:
finite words is well-ordered; for example, the infinite set of words {b, ab, aab, aaab, ... } has no lexicographically earliest element.
322:, including words of length 1 containing a single symbol, words of length 2 with 2 symbols, and so on, even including the empty sequence
4445:
2061:
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.
1060:
Since many applications require well orders, a variant of the lexicographical orders is often used. This well-order, sometimes called
4227:
17:
100:
4258:
72:
2277:
natural numbers. This is not the case for the lexicographical order, as, with the lexicographical order, we have, for example,
3202:
For example, for ordering the increasing sequences (or the sets) of two natural integers, the lexicographical order begins by
1130:
of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a
Cartesian product
4318:
4290:
3351:
As Gröbner bases are defined for polynomials in a fixed number of variables, it is common to identify monomials (for example
4022:
79:
53:
3862:
572:: an earlier CE date is smaller in the lexicographical order than a later date up to year 9999. This date ordering makes
3344:. However this condition is not needed for other related algorithms, such as the algorithms for the computation of the
3293:
2161:
For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of
490:
by the lexicographical order (provided the alphabet is finite); that is, every decreasing sequence of words of length
4362:
1133:
119:
86:
342:
with no symbols at all. The lexicographical order on the set of all these finite words orders the words as follows:
2988:
3354:
520:
531:
of the lexicographic order. In fact, with positional notation, a natural number is represented by a sequence of
68:
57:
3207:
12 < 13 < 14 < 15 < ... < 23 < 24 < 25 < ... < 34 < 35 < ... < 45 < ...
515:
is that it is not always immediately obvious which of two numbers is the smaller. On the other hand, with the
3228:
is finite. In other words, the colexicographical order for increasing sequences of a given length induces an
3345:
1516:, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a
4222:
3541:
3224:
The main property of the colexicographical order for increasing sequences of a given length is that every
453:) at the end until the words are the same length, and then the words are compared as in the previous case.
258:
496:
is finite (or equivalently, every non-empty subset has a least element). It is not true that the set of
4211:
3488:
3237:
2942:
2711:
2676:
2640:
2552:
2518:
2164:
1579:
697:
With this terminology, the above definition of the lexicographical order becomes more concise: Given a
225:; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
3456:
3427:
2893:
2838:
2809:
2611:
2426:
2342:
2021:
1972:
4205:
1673:
621:
508:
The lexicographical order is used not only in dictionaries, but also commonly for numbers and dates.
141:
1847:
4200:
3263:
3258:, the order of the terms does not matter in general, as the addition is commutative. However, some
674:
27:
Generalization of the alphabetical order of dictionaries to sequences of elements of an ordered set
4354:
889:
649:
325:
3267:
2984:
93:
46:
2280:
4183:
1303:
402:, the order of the two words depends on the alphabetic order of the symbols in the first place
2922:
2870:
2789:
1541:
912:
4466:
4461:
3232:
with the natural numbers, and allows enumerating these sequences. This is frequently used in
2743:
2420:
2309:
1642:
1250:
222:
207:
4338:
1085:
4237:
2670:
1199:
512:
4372:
3524:
for distinguishing it from other orders that are also related to a lexicographical order.
2158:. Therefore, in the following, we will consider only orders on subsets of fixed cardinal.
8:
4339:
2228:
123 < 124 < 125 < 126 < 134 < 135 < 136 < 145 < 146 < 156 <
1716:
1677:
569:
554:
516:
184:
2375:
2099:
2076:
1804:
1761:
1275:
1226:
957:
934:
4418:
4334:
2769:
2584:
2398:
2369:
2260:
2126:
2011:
1936:
1916:
1896:
1827:
1784:
1741:
1721:
1698:
1618:
1535:
1495:
1475:
1255:
1179:
984:
745:
under lexicographical order, if at least one of the following conditions is satisfied:
610:(including the empty sequence, of length 0), and the operation (multiplication) is the
172:
137:
133:
604:. That is, the elements of the monoid are the finite sequences (words) of elements of
4358:
4314:
4286:
3229:
2939:
is an order isomorphism when the image is equipped with the lexicographical order on
2254:
1127:
573:
218:
4422:
2233:
234 < 235 < 236 < 245 < 246 < 256 < 345 < 346 < 356 < 456
4408:
4368:
3218:
12 < 13 < 23 < 14 < 24 < 34 < 15 < 25 < 35 < 45 < ...
2015:
1693:
1517:
1108:), and, if the lengths are equal, using the lexicographical order. If the order on
564:
Another example of a non-dictionary use of lexicographical ordering appears in the
543:
3271:
2979:
2549:
The lexicographical ordering may also be used to characterize all group orders on
1781:
They can thus be ordered by the lexicographical order, and for two such functions
4350:
3225:
2250:
1017:
532:
262:
2010:
represented as sets of red squares, increasing sequences (in blue), or by their
4253:
3275:
3266:, require the terms to be in a specific order. Many of the main algorithms for
3249:
2416:
2242:
2155:
2018:(in grey). The grey numbers are also the rank of the subsets in all subsets of
1528:
558:
528:
524:
463:
237:(the set of words used in some language) have a conventional ordering, used in
1824:
the lexicographical order is thus determined by their values for the smallest
1538:
binary sequences (by definition, the set of functions from natural numbers to
4455:
4232:
3233:
2070:
2066:
1521:
698:
611:
458:
242:
195:
3424:
is the number of variables, every monomial order is thus the restriction to
553:(testing the sign takes some time). This is one of the reasons for adopting
4247:
3528:
1639:) does not have a least element under the lexicographical order induced by
1574:
238:
176:
2705:
Robbiano's theorem is that every group order may be obtained in this way.
879:{\displaystyle \varepsilon <b\,\,{\text{ for all }}b\neq \varepsilon ,}
206:
by assigning a total order to the finite set, and converting subsets into
4242:
3279:
2605:
2601:
2453:
2151:
1513:
702:
595:
539:
152:
4440:
4413:
4214:- an algorithmic problem of finding a lexicographically-maximal element.
4182:
A useful property of the degree reverse lexicographical order is that a
4019:
For the lexicographical order, the same exponent vectors are ordered as
1961:
1933:
is finite, then the resulting order is a well-order. As shown above, if
666:) is a prefix of every word, and every word is a prefix of itself (with
408:
where the two words differ (counting from the beginning of the words):
4399:
Weispfenning, Volker (May 1987), "Admissible Orders and Linear Forms",
3255:
1615:) is not well-ordered; the subset of sequences that have precisely one
978:
576:
of dates easier by avoiding the need for a separate sorting algorithm.
487:
249:
203:
2506:{\displaystyle a<b\quad {\text{ if and only if }}\quad a+c<b+c.}
4195:
3287:
3259:
145:
474:
An important property of the lexicographical order is that for each
35:
2150:
In this context, one generally prefer to sort first the subsets by
1062:
565:
550:
180:
2120:
234:
4385:
Robbiano, L. (1985). Term orderings on the polynomial ring. In
3283:
2786:
linear forms with real coefficients, such that the induced map
1966:
199:
132:
For similarly named ordering systems outside mathematics, see
4346:
1070:, consists in considering first the lengths of the words (if
841:
Notice that, due to the prefix condition in this definition,
2253:
are finite, and thus the colexicographical order defines an
3702:{\displaystyle a_{1}+\cdots +a_{n}<b_{1}+\cdots +b_{n},}
2241:
For ordering finite subsets of a given cardinality of the
1114:
is a well-order, the same is true for the shortlex order.
523:, comparing numbers is easy, because the natural order on
4308:
2069:, one has often to enumerate, and therefore to order the
694:); care must be taken if these cases are to be excluded.
4223:
Lexicographic ordering in tensor abstract index notation
2249:
order (see below) is often more convenient, because all
183:
of ordered symbols or, more generally, of elements of a
4259:
Orders on the
Cartesian product of totally ordered sets
2983:
Orderings of the 24 permutations of {1,...,5} that are
1676:. Similarly, the infinite lexicographic product is not
4341:
Information and randomness. An algorithmic perspective
4208:- an application of lexicographic order in economics.
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2714:
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2614:
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2521:
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2429:
2401:
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2345:
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2283:
2263:
2167:
2129:
2102:
2079:
2024:
1975:
1939:
1919:
1899:
1850:
1830:
1807:
1787:
1764:
1744:
1724:
1701:
1645:
1621:
1582:
1544:
1498:
1478:
1332:
1306:
1278:
1258:
1229:
1202:
1182:
1136:
1088:
987:
960:
937:
915:
892:
847:
677:
652:
328:
1687:
2257:between the natural numbers and the set of sets of
1042:has no least element in the lexicographical order:
954:then so is the lexicographic order on the words of
60:. Unsourced material may be challenged and removed.
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4302:
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969:
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898:
878:
686:
658:
346:Given two different words of the same length, say
334:
210:, to which the lexicographical order is applied.
3332:{\displaystyle a<b{\text{ implies }}ac<bc,}
2515:The lexicographical ordering is a group order on
1682:011111... < 101111... < 110111 ... < ...
1122:The lexicographical order defines an order on an
4453:
1670:100000... > 010000... > 001000... > ...
4299:
4280:
4218:Lexicographic order topology on the unit square
1169:{\displaystyle E_{1}\times \cdots \times E_{n}}
4313:. Cambridge University Press. pp. 18–19.
2456:, which is compatible with addition, that is
1298:lexicographical order on the Cartesian product
4250:, a different way of combining partial orders
4398:
4349:Monographs on Theoretical Computer Science.
3527:Another one consists in comparing first the
3411:{\displaystyle x_{1}x_{2}^{3}x_{4}x_{5}^{2}}
2210:
2174:
2043:
2025:
1994:
1976:
1738:may be identified with sequences indexed by
1596:
1583:
1557:
1545:
503:
4389:(pp. 513-517). Springer Berlin Heidelberg.
4276:
4274:
3418:) with their exponent vectors (here ). If
3105:the colexicographical order is defined by
2096:For this, one usually chooses an order on
4412:
4228:Lexicographically minimal string rotation
3494:
3462:
3433:
2948:
2899:
2844:
2815:
2717:
2682:
2646:
2617:
2558:
2524:
2432:
2348:
920:
916:
858:
857:
316:are the finite sequences of symbols from
120:Learn how and when to remove this message
3274:, concept that requires the choice of a
3213:and the colexicographic order begins by
2978:
2974:
1960:
1637:{ 100000..., 010000..., 001000..., ... }
438:in the underlying order of the alphabet
213:A generalization defines an order on an
4446:Lexicographic and colexicographic order
4387:European Conference on Computer Algebra
4271:
2708:More precisely, given a group order on
14:
4454:
4333:
646:. By this definition, the empty word (
4173:{\displaystyle <<<<<.}
2334:
1117:
283:that are not the same symbol, either
4407:(2), New York, NY, USA: ACM: 16–18,
4309:Franz Baader; Tobias Nipkow (1999).
4010:{\displaystyle <<<<<}
3536:degree reverse lexicographical order
3483:
2669:which is injective if the forms are
58:adding citations to reliable sources
29:
24:
3236:, for example in the proof of the
1953:is infinite this is not the case.
1472:The result is a partial order. If
1447:
1426:
1400:
1374:
1361:
1001:is well-ordered. For instance, if
977:However, in general this is not a
624:(or 'truncation') of another word
579:
25:
4478:
4433:
3510:{\displaystyle \mathbb {Z} ^{n},}
2964:{\displaystyle \mathbb {R} ^{s}.}
2733:{\displaystyle \mathbb {Z} ^{n},}
2698:{\displaystyle \mathbb {Z} ^{n}.}
2662:{\displaystyle \mathbb {R} ^{n},}
2574:{\displaystyle \mathbb {Z} ^{n}.}
2540:{\displaystyle \mathbb {Z} ^{n}.}
2216:{\displaystyle S=\{1,2,3,4,5,6\}}
1956:
1688:Functions over a well-ordered set
1608:{\displaystyle \{0,1\}^{\omega }}
4439:
3475:{\displaystyle \mathbb {Z} ^{n}}
3446:{\displaystyle \mathbb {N} ^{n}}
2912:{\displaystyle \mathbb {Z} ^{n}}
2857:{\displaystyle \mathbb {R} ^{s}}
2828:{\displaystyle \mathbb {Z} ^{n}}
2630:{\displaystyle \mathbb {Z} ^{n}}
2608:coefficients, define a map from
2445:{\displaystyle \mathbb {Z} ^{n}}
2395:whose elements are sequences of
2361:{\displaystyle \mathbb {Z} ^{n}}
2052:{\displaystyle \{1,\ldots ,6\},}
2003:{\displaystyle \{1,\ldots ,6\},}
1684:is an infinite ascending chain.
557:representation for representing
248:The formal notion starts with a
194:Another variant, widely used in
34:
3783:
3777:
3290:. Here "compatible" means that
3282:, which is compatible with the
2890:the resulting isomorphism from
2478:
2472:
2415:integers, and operation is the
265:. That is, for any two symbols
45:needs additional citations for
4444:Learning materials related to
4392:
4379:
4327:
4164:
4146:
4140:
4122:
4116:
4098:
4092:
4074:
4068:
4050:
4044:
4026:
4004:
3986:
3980:
3962:
3956:
3938:
3932:
3914:
3908:
3890:
3884:
3866:
3615:
3583:
3577:
3545:
2864:has the following properties;
1884:{\displaystyle f(x)\neq g(x).}
1875:
1869:
1860:
1854:
1345:
1333:
780:(possibly empty) and elements
13:
1:
4264:
687:{\displaystyle =\varepsilon }
480:, the set of words of length
228:
171:) is a generalization of the
4285:. Springer. pp. 88–89.
3243:
2991:(in red) of permutations in
899:{\displaystyle \varepsilon }
659:{\displaystyle \varepsilon }
511:One of the drawbacks of the
335:{\displaystyle \varepsilon }
7:
4311:Term Rewriting and All That
4189:
3809: for the largest
1068:quasi-lexicographical order
735:is non-empty, then one has
527:is the same as the variant
521:Hindu–Arabic numeral system
10:
4483:
4212:Lexicographic optimization
3522:pure lexicographical order
3247:
2475: if and only if
2299:{\displaystyle 12n<134}
1386: if and only if
131:
4206:Lexicographic preferences
1913:is also well-ordered and
1674:infinite descending chain
1319:{\displaystyle A\times B}
504:Numeral systems and dates
142:lexicographic preferences
4281:Egbert Harzheim (2006).
3268:multivariate polynomials
3264:polynomial long division
2932:{\displaystyle \varphi }
2880:{\displaystyle \varphi }
2799:{\displaystyle \varphi }
1566:{\displaystyle \{0,1\},}
1249:Specifically, given two
924:{\displaystyle \,<\,}
18:Lexicographical ordering
3517:for a classification).
3453:of a monomial order of
2759:{\displaystyle s\leq n}
2740:there exist an integer
2325:{\displaystyle n>2.}
1661:{\displaystyle 0<1,}
981:, even if the alphabet
630:if there exists a word
471:alphabetical ordering.
4184:homogeneous polynomial
4174:
4011:
3850:
3703:
3622:
3511:
3484:§ Group orders of
3476:
3447:
3412:
3333:
3238:Kruskal–Katona theorem
3000:
2999:order, and vice versa.
2965:
2933:
2913:
2881:
2858:
2829:
2800:
2780:
2760:
2734:
2699:
2663:
2631:
2595:
2575:
2541:
2507:
2446:
2409:
2389:
2362:
2326:
2300:
2271:
2217:
2137:
2113:
2090:
2062:
2053:
2004:
1947:
1927:
1907:
1885:
1838:
1818:
1795:
1775:
1752:
1732:
1709:
1662:
1629:
1609:
1567:
1506:
1486:
1464:
1320:
1289:
1266:
1251:partially ordered sets
1240:
1217:
1190:
1170:
1102:
1101:{\displaystyle a<b}
995:
971:
948:
925:
900:
880:
688:
660:
336:
223:partially ordered sets
4175:
4012:
3851:
3817: for which
3704:
3623:
3512:
3477:
3448:
3413:
3334:
2982:
2975:Colexicographic order
2966:
2934:
2914:
2882:
2859:
2830:
2801:
2781:
2761:
2735:
2700:
2664:
2632:
2596:
2576:
2542:
2508:
2447:
2410:
2390:
2363:
2327:
2301:
2272:
2218:
2145:lexicographical order
2138:
2114:
2091:
2054:
2005:
1964:
1948:
1928:
1908:
1886:
1839:
1819:
1796:
1776:
1753:
1733:
1710:
1692:The functions from a
1663:
1630:
1610:
1568:
1507:
1487:
1465:
1321:
1290:
1267:
1241:
1218:
1216:{\displaystyle E_{i}}
1191:
1171:
1103:
996:
972:
949:
926:
901:
881:
689:
661:
337:
161:lexicographical order
69:"Lexicographic order"
4238:Long line (topology)
4201:Kleene–Brouwer order
4023:
3863:
3713:
3632:
3621:{\displaystyle <}
3542:
3489:
3457:
3428:
3355:
3294:
2943:
2923:
2894:
2871:
2839:
2810:
2790:
2770:
2744:
2712:
2677:
2671:linearly independent
2641:
2612:
2585:
2553:
2519:
2460:
2427:
2399:
2376:
2343:
2310:
2281:
2261:
2165:
2127:
2100:
2077:
2022:
1973:
1937:
1917:
1897:
1848:
1828:
1805:
1785:
1762:
1742:
1722:
1699:
1643:
1619:
1580:
1542:
1496:
1476:
1330:
1304:
1276:
1256:
1227:
1200:
1180:
1176:is a sequence whose
1134:
1086:
985:
958:
935:
931:is a total order on
913:
890:
845:
675:
650:
574:computerized sorting
513:Roman numeral system
326:
208:increasing sequences
54:improve this article
4414:10.1145/24554.24557
3407:
3382:
3308: implies
2012:indicator functions
1965:Orderings of the 3-
1717:totally ordered set
1196:element belongs to
906:is the empty word.
861: for all
762:there exists words
570:chronological order
517:positional notation
257:, often called the
185:totally ordered set
4170:
4007:
3846:
3699:
3618:
3507:
3472:
3443:
3408:
3393:
3368:
3329:
3001:
2961:
2929:
2909:
2877:
2854:
2825:
2796:
2776:
2756:
2730:
2695:
2659:
2627:
2591:
2571:
2537:
2503:
2442:
2405:
2388:{\displaystyle n,}
2385:
2370:free Abelian group
2358:
2322:
2296:
2267:
2213:
2133:
2112:{\displaystyle S.}
2109:
2089:{\displaystyle S.}
2086:
2063:
2049:
2000:
1943:
1923:
1903:
1881:
1834:
1817:{\displaystyle g,}
1814:
1791:
1774:{\displaystyle Y.}
1771:
1748:
1728:
1705:
1658:
1625:
1605:
1573:also known as the
1563:
1536:countably infinite
1502:
1482:
1460:
1316:
1288:{\displaystyle B,}
1285:
1262:
1239:{\displaystyle i.}
1236:
1213:
1186:
1166:
1118:Cartesian products
1098:
991:
970:{\displaystyle A.}
967:
947:{\displaystyle A,}
944:
921:
896:
876:
684:
656:
332:
173:alphabetical order
138:natural sort order
134:alphabetical order
4320:978-0-521-77920-3
4292:978-0-387-24222-4
3818:
3810:
3781:
3309:
3286:structure of the
3270:are related with
3254:When considering
3230:order isomorphism
2989:inversion vectors
2779:{\displaystyle s}
2594:{\displaystyle n}
2476:
2408:{\displaystyle n}
2335:Group orders of Z
2270:{\displaystyle n}
2255:order isomorphism
2247:colexicographical
2154:, such as in the
2136:{\displaystyle S}
1946:{\displaystyle X}
1926:{\displaystyle X}
1906:{\displaystyle Y}
1837:{\displaystyle x}
1794:{\displaystyle f}
1751:{\displaystyle X}
1731:{\displaystyle Y}
1708:{\displaystyle X}
1628:{\displaystyle 1}
1505:{\displaystyle B}
1485:{\displaystyle A}
1434:
1408:
1387:
1265:{\displaystyle A}
1189:{\displaystyle i}
1128:Cartesian product
994:{\displaystyle A}
862:
614:of words. A word
588:over an alphabet
219:Cartesian product
130:
129:
122:
104:
16:(Redirected from
4474:
4443:
4427:
4425:
4416:
4396:
4390:
4383:
4377:
4376:
4344:
4335:Calude, Cristian
4331:
4325:
4324:
4306:
4297:
4296:
4278:
4179:
4177:
4176:
4171:
4016:
4014:
4013:
4008:
3855:
3853:
3852:
3847:
3842:
3841:
3829:
3828:
3819:
3816:
3811:
3808:
3806:
3805:
3793:
3792:
3782:
3779:
3776:
3775:
3757:
3756:
3744:
3743:
3725:
3724:
3708:
3706:
3705:
3700:
3695:
3694:
3676:
3675:
3663:
3662:
3644:
3643:
3627:
3625:
3624:
3619:
3614:
3613:
3595:
3594:
3576:
3575:
3557:
3556:
3516:
3514:
3513:
3508:
3503:
3502:
3497:
3481:
3479:
3478:
3473:
3471:
3470:
3465:
3452:
3450:
3449:
3444:
3442:
3441:
3436:
3423:
3417:
3415:
3414:
3409:
3406:
3401:
3392:
3391:
3381:
3376:
3367:
3366:
3343:
3338:
3336:
3335:
3330:
3310:
3307:
3219:
3208:
3193:
3184:
3175:
3169:
3153:
3100:
3091:
3082:
3076:
3060:
2970:
2968:
2967:
2962:
2957:
2956:
2951:
2938:
2936:
2935:
2930:
2919:to the image of
2918:
2916:
2915:
2910:
2908:
2907:
2902:
2886:
2884:
2883:
2878:
2863:
2861:
2860:
2855:
2853:
2852:
2847:
2834:
2832:
2831:
2826:
2824:
2823:
2818:
2805:
2803:
2802:
2797:
2785:
2783:
2782:
2777:
2765:
2763:
2762:
2757:
2739:
2737:
2736:
2731:
2726:
2725:
2720:
2704:
2702:
2701:
2696:
2691:
2690:
2685:
2668:
2666:
2665:
2660:
2655:
2654:
2649:
2636:
2634:
2633:
2628:
2626:
2625:
2620:
2600:
2598:
2597:
2592:
2580:
2578:
2577:
2572:
2567:
2566:
2561:
2546:
2544:
2543:
2538:
2533:
2532:
2527:
2512:
2510:
2509:
2504:
2477:
2474:
2451:
2449:
2448:
2443:
2441:
2440:
2435:
2414:
2412:
2411:
2406:
2394:
2392:
2391:
2386:
2367:
2365:
2364:
2359:
2357:
2356:
2351:
2331:
2329:
2328:
2323:
2305:
2303:
2302:
2297:
2276:
2274:
2273:
2268:
2251:initial segments
2234:
2229:
2222:
2220:
2219:
2214:
2142:
2140:
2139:
2134:
2118:
2116:
2115:
2110:
2095:
2093:
2092:
2087:
2058:
2056:
2055:
2050:
2016:decimal notation
2009:
2007:
2006:
2001:
1952:
1950:
1949:
1944:
1932:
1930:
1929:
1924:
1912:
1910:
1909:
1904:
1890:
1888:
1887:
1882:
1843:
1841:
1840:
1835:
1823:
1821:
1820:
1815:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1714:
1712:
1711:
1706:
1694:well-ordered set
1683:
1671:
1667:
1665:
1664:
1659:
1638:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1604:
1603:
1572:
1570:
1569:
1564:
1518:linear extension
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1469:
1467:
1466:
1461:
1456:
1452:
1451:
1450:
1435:
1432:
1430:
1429:
1409:
1406:
1404:
1403:
1388:
1385:
1383:
1379:
1378:
1377:
1365:
1364:
1325:
1323:
1322:
1317:
1300:
1299:
1294:
1292:
1291:
1286:
1271:
1269:
1268:
1263:
1245:
1243:
1242:
1237:
1222:
1220:
1219:
1214:
1212:
1211:
1195:
1193:
1192:
1187:
1175:
1173:
1172:
1167:
1165:
1164:
1146:
1145:
1113:
1107:
1105:
1104:
1099:
1081:
1056:
1041:
1015:
1000:
998:
997:
992:
976:
974:
973:
968:
953:
951:
950:
945:
930:
928:
927:
922:
905:
903:
902:
897:
885:
883:
882:
877:
863:
860:
835:
824:
813:
797:
791:
785:
779:
773:
767:
759:
753:
744:
734:
728:
722:
716:
711:, and two words
710:
693:
691:
690:
685:
671:
665:
663:
662:
657:
645:
635:
629:
619:
609:
603:
593:
555:two's complement
544:decimal notation
533:numerical digits
495:
485:
479:
452:
443:
437:
417:
407:
401:
373:
341:
339:
338:
333:
321:
315:
302:
292:
282:
276:
270:
256:
169:dictionary order
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
4482:
4481:
4477:
4476:
4475:
4473:
4472:
4471:
4452:
4451:
4436:
4431:
4430:
4401:SIGSAM Bulletin
4397:
4393:
4384:
4380:
4365:
4351:Springer-Verlag
4332:
4328:
4321:
4307:
4300:
4293:
4279:
4272:
4267:
4192:
4024:
4021:
4020:
3864:
3861:
3860:
3837:
3833:
3824:
3820:
3815:
3807:
3801:
3797:
3788:
3784:
3780: and
3778:
3771:
3767:
3752:
3748:
3739:
3735:
3720:
3716:
3714:
3711:
3710:
3690:
3686:
3671:
3667:
3658:
3654:
3639:
3635:
3633:
3630:
3629:
3609:
3605:
3590:
3586:
3571:
3567:
3552:
3548:
3543:
3540:
3539:
3498:
3493:
3492:
3490:
3487:
3486:
3466:
3461:
3460:
3458:
3455:
3454:
3437:
3432:
3431:
3429:
3426:
3425:
3419:
3402:
3397:
3387:
3383:
3377:
3372:
3362:
3358:
3356:
3353:
3352:
3341:
3306:
3295:
3292:
3291:
3252:
3246:
3226:initial segment
3217:
3206:
3191:
3186:
3182:
3177:
3171:
3167:
3160:
3155:
3152:
3143:
3137:
3130:
3121:
3115:
3109:
3098:
3093:
3089:
3084:
3078:
3074:
3067:
3062:
3059:
3050:
3044:
3037:
3028:
3022:
3016:
3005:colexicographic
2987:(in blue). The
2977:
2952:
2947:
2946:
2944:
2941:
2940:
2924:
2921:
2920:
2903:
2898:
2897:
2895:
2892:
2891:
2872:
2869:
2868:
2848:
2843:
2842:
2840:
2837:
2836:
2819:
2814:
2813:
2811:
2808:
2807:
2791:
2788:
2787:
2771:
2768:
2767:
2745:
2742:
2741:
2721:
2716:
2715:
2713:
2710:
2709:
2686:
2681:
2680:
2678:
2675:
2674:
2650:
2645:
2644:
2642:
2639:
2638:
2621:
2616:
2615:
2613:
2610:
2609:
2586:
2583:
2582:
2562:
2557:
2556:
2554:
2551:
2550:
2528:
2523:
2522:
2520:
2517:
2516:
2473:
2461:
2458:
2457:
2436:
2431:
2430:
2428:
2425:
2424:
2400:
2397:
2396:
2377:
2374:
2373:
2352:
2347:
2346:
2344:
2341:
2340:
2337:
2311:
2308:
2307:
2282:
2279:
2278:
2262:
2259:
2258:
2243:natural numbers
2232:
2227:
2166:
2163:
2162:
2128:
2125:
2124:
2101:
2098:
2097:
2078:
2075:
2074:
2073:of a given set
2060:
2023:
2020:
2019:
2014:, converted in
1974:
1971:
1970:
1959:
1938:
1935:
1934:
1918:
1915:
1914:
1898:
1895:
1894:
1849:
1846:
1845:
1829:
1826:
1825:
1806:
1803:
1802:
1786:
1783:
1782:
1763:
1760:
1759:
1758:of elements of
1743:
1740:
1739:
1723:
1720:
1719:
1700:
1697:
1696:
1690:
1681:
1680:either because
1669:
1644:
1641:
1640:
1636:
1620:
1617:
1616:
1599:
1595:
1581:
1578:
1577:
1543:
1540:
1539:
1529:natural numbers
1514:totally ordered
1497:
1494:
1493:
1477:
1474:
1473:
1446:
1442:
1433: and
1431:
1425:
1421:
1414:
1410:
1405:
1399:
1395:
1384:
1373:
1369:
1360:
1356:
1355:
1351:
1331:
1328:
1327:
1305:
1302:
1301:
1297:
1296:
1277:
1274:
1273:
1257:
1254:
1253:
1228:
1225:
1224:
1207:
1203:
1201:
1198:
1197:
1181:
1178:
1177:
1160:
1156:
1141:
1137:
1135:
1132:
1131:
1120:
1109:
1087:
1084:
1083:
1071:
1043:
1020:
1002:
986:
983:
982:
959:
956:
955:
936:
933:
932:
914:
911:
910:
891:
888:
887:
859:
846:
843:
842:
827:
816:
805:
793:
787:
781:
775:
769:
763:
755:
754:is a prefix of
749:
736:
730:
724:
718:
712:
706:
703:totally ordered
676:
673:
672:
667:
651:
648:
647:
637:
631:
625:
615:
605:
599:
589:
586:monoid of words
582:
580:Monoid of words
559:signed integers
525:natural numbers
506:
491:
481:
475:
448:
439:
436:
427:
419:
418:if and only if
409:
403:
400:
391:
385:
375:
372:
363:
357:
347:
327:
324:
323:
317:
311:
294:
284:
278:
272:
266:
263:totally ordered
252:
233:The words in a
231:
163:(also known as
149:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4480:
4470:
4469:
4464:
4450:
4449:
4448:at Wikiversity
4435:
4434:External links
4432:
4429:
4428:
4391:
4378:
4363:
4326:
4319:
4298:
4291:
4269:
4268:
4266:
4263:
4262:
4261:
4256:
4254:Shortlex order
4251:
4245:
4240:
4235:
4230:
4225:
4220:
4215:
4209:
4203:
4198:
4191:
4188:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3845:
3840:
3836:
3832:
3827:
3823:
3814:
3804:
3800:
3796:
3791:
3787:
3774:
3770:
3766:
3763:
3760:
3755:
3751:
3747:
3742:
3738:
3734:
3731:
3728:
3723:
3719:
3698:
3693:
3689:
3685:
3682:
3679:
3674:
3670:
3666:
3661:
3657:
3653:
3650:
3647:
3642:
3638:
3617:
3612:
3608:
3604:
3601:
3598:
3593:
3589:
3585:
3582:
3579:
3574:
3570:
3566:
3563:
3560:
3555:
3551:
3547:
3537:
3523:
3506:
3501:
3496:
3469:
3464:
3440:
3435:
3405:
3400:
3396:
3390:
3386:
3380:
3375:
3371:
3365:
3361:
3328:
3325:
3322:
3319:
3316:
3313:
3305:
3302:
3299:
3276:monomial order
3250:Monomial order
3248:Main article:
3245:
3242:
3222:
3221:
3211:
3210:
3196:
3195:
3189:
3180:
3165:
3158:
3148:
3141:
3135:
3126:
3119:
3113:
3103:
3102:
3096:
3087:
3077:for the first
3072:
3065:
3055:
3048:
3042:
3033:
3026:
3020:
2976:
2973:
2972:
2971:
2960:
2955:
2950:
2928:
2906:
2901:
2888:
2876:
2851:
2846:
2822:
2817:
2795:
2775:
2755:
2752:
2749:
2729:
2724:
2719:
2694:
2689:
2684:
2658:
2653:
2648:
2624:
2619:
2590:
2570:
2565:
2560:
2536:
2531:
2526:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2471:
2468:
2465:
2439:
2434:
2404:
2384:
2381:
2355:
2350:
2336:
2333:
2321:
2318:
2315:
2295:
2292:
2289:
2286:
2266:
2248:
2239:
2238:
2237:
2236:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2156:shortlex order
2146:
2132:
2108:
2105:
2085:
2082:
2071:finite subsets
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1958:
1957:Finite subsets
1955:
1942:
1922:
1902:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1833:
1813:
1810:
1790:
1770:
1767:
1747:
1727:
1704:
1689:
1686:
1657:
1654:
1651:
1648:
1624:
1602:
1598:
1594:
1591:
1588:
1585:
1562:
1559:
1556:
1553:
1550:
1547:
1501:
1481:
1459:
1455:
1449:
1445:
1441:
1438:
1428:
1424:
1420:
1417:
1413:
1407: or
1402:
1398:
1394:
1391:
1382:
1376:
1372:
1368:
1363:
1359:
1354:
1350:
1347:
1344:
1341:
1338:
1335:
1326:is defined as
1315:
1312:
1309:
1284:
1281:
1261:
1235:
1232:
1210:
1206:
1185:
1163:
1159:
1155:
1152:
1149:
1144:
1140:
1119:
1116:
1097:
1094:
1091:
1076:) < length(
1069:
1065:
990:
966:
963:
943:
940:
919:
895:
875:
872:
869:
866:
856:
853:
850:
839:
838:
837:
836:
825:
814:
800:
799:
760:
683:
680:
655:
587:
581:
578:
561:in computers.
505:
502:
466:
464:shortlex order
455:
454:
445:
432:
423:
396:
389:
383:
368:
361:
355:
331:
230:
227:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
4479:
4468:
4465:
4463:
4460:
4459:
4457:
4447:
4442:
4438:
4437:
4424:
4420:
4415:
4410:
4406:
4402:
4395:
4388:
4382:
4374:
4370:
4366:
4364:3-540-57456-5
4360:
4356:
4352:
4348:
4343:
4342:
4336:
4330:
4322:
4316:
4312:
4305:
4303:
4294:
4288:
4284:
4277:
4275:
4270:
4260:
4257:
4255:
4252:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4233:Leximin order
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4213:
4210:
4207:
4204:
4202:
4199:
4197:
4194:
4193:
4187:
4185:
4180:
4167:
4161:
4158:
4155:
4152:
4149:
4143:
4137:
4134:
4131:
4128:
4125:
4119:
4113:
4110:
4107:
4104:
4101:
4095:
4089:
4086:
4083:
4080:
4077:
4071:
4065:
4062:
4059:
4056:
4053:
4047:
4041:
4038:
4035:
4032:
4029:
4017:
4001:
3998:
3995:
3992:
3989:
3983:
3977:
3974:
3971:
3968:
3965:
3959:
3953:
3950:
3947:
3944:
3941:
3935:
3929:
3926:
3923:
3920:
3917:
3911:
3905:
3902:
3899:
3896:
3893:
3887:
3881:
3878:
3875:
3872:
3869:
3856:
3843:
3838:
3834:
3830:
3825:
3821:
3812:
3802:
3798:
3794:
3789:
3785:
3772:
3768:
3764:
3761:
3758:
3753:
3749:
3745:
3740:
3736:
3732:
3729:
3726:
3721:
3717:
3696:
3691:
3687:
3683:
3680:
3677:
3672:
3668:
3664:
3659:
3655:
3651:
3648:
3645:
3640:
3636:
3610:
3606:
3602:
3599:
3596:
3591:
3587:
3580:
3572:
3568:
3564:
3561:
3558:
3553:
3549:
3535:
3532:
3530:
3529:total degrees
3525:
3521:
3518:
3504:
3499:
3485:
3467:
3438:
3422:
3403:
3398:
3394:
3388:
3384:
3378:
3373:
3369:
3363:
3359:
3349:
3347:
3326:
3323:
3320:
3317:
3314:
3311:
3303:
3300:
3297:
3289:
3285:
3281:
3277:
3273:
3272:Gröbner bases
3269:
3265:
3261:
3257:
3251:
3241:
3239:
3235:
3234:combinatorics
3231:
3227:
3216:
3215:
3214:
3205:
3204:
3203:
3200:
3192:
3183:
3174:
3170:for the last
3168:
3161:
3151:
3147:
3140:
3134:
3129:
3125:
3118:
3112:
3108:
3107:
3106:
3099:
3090:
3081:
3075:
3068:
3058:
3054:
3047:
3041:
3036:
3032:
3025:
3019:
3015:
3014:
3013:
3010:
3006:
2998:
2995:order are in
2994:
2990:
2986:
2981:
2958:
2953:
2926:
2904:
2889:
2887:is injective;
2874:
2867:
2866:
2865:
2849:
2820:
2793:
2773:
2753:
2750:
2747:
2727:
2722:
2706:
2692:
2687:
2672:
2656:
2651:
2622:
2607:
2603:
2588:
2568:
2563:
2547:
2534:
2529:
2513:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2469:
2466:
2463:
2455:
2437:
2422:
2418:
2402:
2382:
2379:
2371:
2353:
2332:
2319:
2316:
2313:
2293:
2290:
2287:
2284:
2264:
2256:
2252:
2246:
2244:
2231:
2230:
2226:
2225:
2224:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2171:
2168:
2159:
2157:
2153:
2148:
2144:
2130:
2122:
2106:
2103:
2083:
2080:
2072:
2068:
2067:combinatorics
2046:
2040:
2037:
2034:
2031:
2028:
2017:
2013:
1997:
1991:
1988:
1985:
1982:
1979:
1968:
1963:
1954:
1940:
1920:
1900:
1891:
1878:
1872:
1866:
1863:
1857:
1851:
1831:
1811:
1808:
1788:
1768:
1765:
1745:
1725:
1718:
1702:
1695:
1685:
1679:
1675:
1655:
1652:
1649:
1646:
1622:
1600:
1592:
1589:
1586:
1576:
1560:
1554:
1551:
1548:
1537:
1532:
1530:
1525:
1523:
1522:product order
1519:
1515:
1499:
1479:
1470:
1457:
1453:
1443:
1439:
1436:
1422:
1418:
1415:
1411:
1396:
1392:
1389:
1380:
1370:
1366:
1357:
1352:
1348:
1342:
1339:
1336:
1313:
1310:
1307:
1282:
1279:
1259:
1252:
1247:
1233:
1230:
1208:
1204:
1183:
1161:
1157:
1153:
1150:
1147:
1142:
1138:
1129:
1125:
1115:
1112:
1095:
1092:
1089:
1079:
1075:
1067:
1064:
1061:
1058:
1055:
1051:
1047:
1039:
1035:
1031:
1027:
1024:
1019:
1013:
1009:
1005:
988:
980:
964:
961:
941:
938:
917:
907:
893:
873:
870:
867:
864:
854:
851:
848:
834:
830:
826:
823:
819:
815:
812:
808:
804:
803:
802:
801:
796:
790:
784:
778:
772:
766:
761:
758:
752:
748:
747:
746:
743:
739:
733:
727:
721:
715:
709:
704:
700:
695:
681:
678:
670:
653:
644:
640:
634:
628:
623:
618:
613:
612:concatenation
608:
602:
597:
592:
585:
577:
575:
571:
567:
562:
560:
556:
552:
547:
545:
541:
536:
534:
530:
526:
522:
518:
514:
509:
501:
499:
494:
489:
484:
478:
472:
468:
465:
462:
460:
459:combinatorics
451:
446:
442:
435:
431:
426:
422:
416:
412:
406:
399:
395:
388:
382:
378:
371:
367:
360:
354:
350:
345:
344:
343:
329:
320:
314:
309:
304:
301:
297:
291:
287:
281:
275:
269:
264:
260:
255:
251:
246:
244:
243:encyclopedias
240:
236:
226:
224:
220:
216:
211:
209:
205:
201:
197:
196:combinatorics
192:
188:
186:
182:
178:
174:
170:
166:
165:lexical order
162:
158:
157:lexicographic
154:
147:
143:
139:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4467:Lexicography
4462:Order theory
4404:
4400:
4394:
4386:
4381:
4340:
4329:
4310:
4283:Ordered Sets
4282:
4248:Star product
4181:
4018:
3857:
3533:
3526:
3519:
3420:
3350:
3346:tangent cone
3278:, that is a
3253:
3223:
3212:
3201:
3197:
3187:
3178:
3172:
3163:
3156:
3149:
3145:
3138:
3132:
3127:
3123:
3116:
3110:
3104:
3094:
3085:
3079:
3070:
3063:
3056:
3052:
3045:
3039:
3034:
3030:
3023:
3017:
3008:
3004:
3002:
2996:
2992:
2707:
2602:linear forms
2548:
2514:
2338:
2240:
2160:
2149:
2123:a subset of
2064:
1892:
1691:
1575:Cantor space
1533:
1526:
1471:
1248:
1123:
1121:
1110:
1077:
1073:
1059:
1053:
1049:
1045:
1037:
1033:
1029:
1025:
1022:
1011:
1007:
1003:
908:
840:
832:
828:
821:
817:
810:
806:
794:
788:
782:
776:
770:
764:
756:
750:
741:
737:
731:
725:
719:
713:
707:
696:
668:
642:
638:
632:
626:
616:
606:
600:
590:
583:
563:
548:
540:real numbers
537:
510:
507:
497:
492:
488:well-ordered
482:
476:
473:
469:
457:However, in
456:
449:
440:
433:
429:
424:
420:
414:
410:
404:
397:
393:
386:
380:
376:
369:
365:
358:
352:
348:
318:
312:
307:
305:
299:
295:
289:
285:
279:
273:
267:
253:
247:
239:dictionaries
232:
214:
212:
193:
189:
177:dictionaries
168:
164:
160:
156:
150:
116:
110:January 2022
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
4243:Lyndon word
3628:if either
3482:(see above
3280:total order
3256:polynomials
3009:colex order
2454:total order
2421:group order
2152:cardinality
596:free monoid
542:written in
261:, which is
202:of a given
153:mathematics
4456:Categories
4373:0922.68073
4353:. p.
4265:References
3262:, such as
3260:algorithms
2306:for every
1844:such that
1678:Noetherian
1635:(that is,
1223:for every
979:well-order
729:such that
636:such that
250:finite set
229:Definition
204:finite set
80:newspapers
4196:Collation
3831:≠
3762:⋯
3730:⋯
3681:⋯
3649:⋯
3600:…
3562:…
3288:monomials
3244:Monomials
2927:φ
2875:φ
2794:φ
2751:≤
2581:In fact,
2035:…
1986:…
1864:≠
1601:ω
1520:of their
1512:are each
1448:′
1440:≤
1427:′
1401:′
1375:′
1362:′
1349:≤
1311:×
1154:×
1151:⋯
1148:×
1044:... <
894:ε
871:ε
868:≠
849:ε
798:such that
699:partially
682:ε
654:ε
551:computers
330:ε
198:, orders
181:sequences
146:collation
4423:10226875
4337:(1994).
4190:See also
2997:revcolex
2985:5-cycles
2417:addition
2372:of rank
1668:because
1063:shortlex
1018:language
566:ISO 8601
529:shortlex
259:alphabet
3101:differ,
2368:be the
2121:sorting
1967:subsets
1082:, then
1072:length(
594:is the
519:of the
235:lexicon
200:subsets
175:of the
94:scholar
4421:
4371:
4361:
4317:
4289:
3284:monoid
3194:differ
3176:where
3083:where
2245:, the
2119:Then,
1672:is an
1016:, the
886:where
622:prefix
155:, the
144:, and
96:
89:
82:
75:
67:
4419:S2CID
4347:EATCS
3162:<
3131:<
3069:<
3038:<
2993:colex
2835:into
2806:from
2637:into
2604:with
2452:is a
1715:to a
1126:-ary
1052:<
1048:<
1036:>
1032:≥ 0,
809:<
740:<
723:over
620:is a
598:over
428:<
413:<
308:words
298:<
288:<
217:-ary
167:, or
101:JSTOR
87:books
4359:ISBN
4315:ISBN
4287:ISBN
4144:<
4120:<
4096:<
4072:<
4048:<
3984:<
3960:<
3936:<
3912:<
3888:<
3795:>
3665:<
3581:<
3534:The
3318:<
3301:<
3185:and
3092:and
3051:...
3003:The
2766:and
2606:real
2489:<
2467:<
2419:. A
2339:Let
2317:>
2291:<
1801:and
1650:<
1492:and
1393:<
1295:the
1272:and
1093:<
918:<
852:<
786:and
717:and
705:set
584:The
538:For
374:and
306:The
271:and
241:and
73:news
4409:doi
4369:Zbl
3709:or
3154:if
3144:...
3122:...
3061:if
3029:...
3007:or
2423:on
2294:134
2223:is
2065:In
1969:of
1893:If
1066:or
1046:aab
1006:= {
909:If
833:uyw
822:uxv
792:of
701:or
498:all
486:is
392:...
364:...
310:of
293:or
277:in
221:of
179:to
159:or
151:In
56:by
4458::
4417:,
4405:21
4403:,
4367:.
4357:.
4345:.
4301:^
4273:^
3348:.
3240:.
2320:2.
2285:12
2147:.
1524:.
1057:.
1050:ab
1028:|
1010:,
831:=
820:=
774:,
768:,
643:uw
641:=
467:.
379:=
351:=
303:.
187:.
140:,
136:,
4426:.
4411::
4375:.
4355:1
4323:.
4295:.
4168:.
4165:]
4162:0
4159:,
4156:0
4153:,
4150:2
4147:[
4141:]
4138:0
4135:,
4132:1
4129:,
4126:1
4123:[
4117:]
4114:1
4111:,
4108:0
4105:,
4102:1
4099:[
4093:]
4090:0
4087:,
4084:2
4081:,
4078:0
4075:[
4069:]
4066:1
4063:,
4060:1
4057:,
4054:0
4051:[
4045:]
4042:2
4039:,
4036:0
4033:,
4030:0
4027:[
4005:]
4002:0
3999:,
3996:0
3993:,
3990:2
3987:[
3981:]
3978:0
3975:,
3972:1
3969:,
3966:1
3963:[
3957:]
3954:0
3951:,
3948:2
3945:,
3942:0
3939:[
3933:]
3930:1
3927:,
3924:0
3921:,
3918:1
3915:[
3909:]
3906:1
3903:,
3900:1
3897:,
3894:0
3891:[
3885:]
3882:2
3879:,
3876:0
3873:,
3870:0
3867:[
3844:.
3839:i
3835:b
3826:i
3822:a
3813:i
3803:i
3799:b
3790:i
3786:a
3773:n
3769:b
3765:+
3759:+
3754:1
3750:b
3746:=
3741:n
3737:a
3733:+
3727:+
3722:1
3718:a
3697:,
3692:n
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3584:[
3578:]
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2728:,
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2589:n
2569:.
2564:n
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2535:.
2530:n
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2501:.
2498:c
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2235:.
2211:}
2208:6
2205:,
2202:5
2199:,
2196:4
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2175:{
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2169:S
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1858:x
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1584:{
1561:,
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1500:B
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679:=
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639:v
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268:a
254:A
215:n
148:.
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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