1642:
1629:
1063:
1053:
1023:
1013:
991:
981:
946:
924:
914:
879:
847:
837:
812:
780:
765:
745:
713:
703:
693:
673:
641:
621:
611:
601:
569:
549:
505:
485:
475:
438:
413:
403:
371:
346:
309:
279:
247:
185:
150:
2019:) of a finite set is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite
2710:) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals. However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that
3407:
Subsets that are unbounded by themselves but bounded in the whole set; they have no maximum, but a supremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hence also of the whole set; if the subset is empty this supremum is the minimum of the whole
2154:
2363:
2613:
1373:
1302:
2522:
1617:
1231:
1502:
1458:
1414:
2015:, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements,
3192:
3278:
1639:
indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by
1572:
1469:
1425:
1384:
1313:
1242:
1186:
3204:
within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points.
3096:
2076:
3023:
2857:
2286:
2070:
Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:
2533:
1175:
1561:
2690:
3389:
2892:
2930:
2743:
2418:
1308:
2011:
of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of
1715:
1237:
1796:
1767:
1741:
1529:
1146:
1676:
3404:
by themselves); this can be an isolated point or a limit point of the whole set; in the latter case it may or may not be also a limit point of the subset.
3357:— this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type
3496:
Bonnet, Rémi; Finkel, Alain; Haddad, Serge; Rosa-Velardo, Fernando (2013). "Ordinal theory for expressiveness of well-structured transition systems".
2428:
1567:
1181:
3291:
of this set, 1 is a limit point of the set, despite being separated from the only limit point 0 under the ordinary topology of the real numbers.
1464:
1420:
1379:
4269:
50:
4252:
3108:
3210:
3782:
3618:
3586:
2859:
whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an
3419:
in the whole set if and only if it is unbounded in the whole set or it has a maximum that is also maximum of the whole set.
4099:
2942:
is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard
2149:{\displaystyle {\begin{matrix}0&2&4&6&8&\dots &1&3&5&7&9&\dots \end{matrix}}}
3040:
3317:
Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the
4235:
4094:
1943:. The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible.
4089:
2707:
3725:
2027:-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "
2960:
2358:{\displaystyle {\begin{matrix}0&1&2&3&4&\dots &-1&-2&-3&\dots \end{matrix}}}
2067:
is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.
1936:
well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a
3807:
2793:
43:
4126:
4046:
2042:, but not conversely: well-ordered sets of a particular cardinality can have many different order types (see
1897:, has a unique successor (next element), namely the least element of the subset of all elements greater than
2608:{\displaystyle {\begin{matrix}0&-1&1&-2&2&-3&3&-4&4&\dots \end{matrix}}}
3911:
3840:
3720:
2172:. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1.
3814:
3802:
3765:
3740:
3715:
3669:
3638:
1151:
4111:
3745:
3735:
3611:
2717:
An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose
2934:
to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from
1535:
4084:
3750:
3431:
3401:
3318:
2645:
1966:, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a
36:
4312:
4307:
4016:
3643:
3369:
2872:
1110:
934:
103:
2913:
2726:
2714:, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
1368:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
4264:
4247:
3540:
2390:
1982:
4176:
3792:
2703:
2634:
1297:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1080:
73:
4302:
4154:
3989:
3980:
3849:
3730:
3684:
3648:
3604:
3423:
3416:
3311:
2908:, which could be mapped to zero later). And it is well known that there is an injection from
1971:
1967:
1959:
1937:
1635:
indicates that the column's property is always true the row's term (at the very left), while
1090:
83:
1685:
4242:
4201:
4191:
4181:
3926:
3889:
3879:
3859:
3844:
3578:
3517:
1656:
867:
142:
1772:
8:
4169:
4080:
4026:
3985:
3975:
3864:
3797:
3760:
1799:
A term's definition may require additional properties that are not listed in this table.
1746:
1720:
1679:
1508:
1125:
1115:
526:
108:
25:
3572:
4281:
4208:
4061:
3970:
3960:
3901:
3819:
3755:
3467:
2860:
2047:
1940:
1661:
1105:
1085:
1075:
1001:
98:
78:
68:
4121:
4218:
4196:
4056:
4041:
4021:
3824:
3582:
3451:
3334:
1918:
1825:
4031:
3884:
3549:
3531:
3505:
3462:
2000:
1947:
1933:
1894:
800:
733:
1901:. There may be elements, besides the least element, that have no predecessor (see
4213:
3996:
3874:
3869:
3854:
3770:
3679:
3664:
3513:
3472:
3362:
2699:
2196:
2189:
2016:
1978:
1974:
can be used to prove that a given statement is true for all elements of the set.
1963:
661:
458:
29:
2517:{\displaystyle |x|<|y|\qquad {\text{or}}\qquad |x|=|y|{\text{ and }}x\leq y.}
4131:
4116:
4106:
3965:
3943:
3921:
3568:
3457:
3348:
3338:
3288:
3025:
has no least element and is therefore not a well order under standard ordering
2420:
2372:
2206:
2181:
2064:
2004:
1994:
1951:
1832:
1612:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
329:
1226:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
4296:
4230:
4186:
4164:
4036:
3906:
3894:
3699:
3509:
2638:
2387:
Another relation for well ordering the integers is the following definition:
1862:. In some academic articles and textbooks these terms are instead written as
1851:
1100:
1095:
267:
93:
88:
3554:
3308:
The set is well ordered. That is, every nonempty subset has a least element.
4051:
3933:
3916:
3834:
3674:
3627:
3477:
4257:
3950:
3829:
3694:
3439:
3354:
3301:
3201:
2039:
1914:
1809:
391:
1981:
are well ordered by the usual less-than relation is commonly called the
1497:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
4225:
4159:
2619:
2160:
2008:
1955:
1453:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
4276:
4149:
3955:
3435:
1840:
1409:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2035:
can also be an infinite ordinal, it will typically count from zero.
1889:
Every non-empty well-ordered set has a least element. Every element
4071:
3938:
3689:
2012:
206:
3344:
With respect to this topology there can be two kinds of elements:
2950:
The natural numbers are a well order under the standard ordering
2185:
1921:, namely the least element of the subset of all upper bounds of
3596:
3535:
1843:
3536:"Some Applications of the Notions of Forcing and Generic Sets"
3495:
3187:{\displaystyle \{-2^{-n}-2^{-m-n}\,|\,0\leq m,n<\omega \}}
2702:) one can show that there is a well order of the reals. Also
3351:— these are the minimum and the elements with a predecessor.
3273:{\displaystyle \{-2^{-n}\,|\,0\leq n<\omega \}\cup \{1\}}
2711:
2695:
3426:
if and only if it has order type less than or equal to ω
3574:
Real
Analysis: Modern Techniques and Their Applications
2188:
is not a well ordering, since, for example, the set of
3324:
Every subordering is isomorphic to an initial segment.
2538:
2291:
2081:
3372:
3213:
3111:
3043:
2963:
2916:
2875:
2796:
2729:
2648:
2536:
2431:
2393:
2289:
2079:
2050:
set, the set of possible order types is uncountable.
1775:
1749:
1723:
1688:
1664:
1570:
1538:
1511:
1467:
1423:
1382:
1311:
1240:
1184:
1154:
1128:
3304:, then the following are equivalent to each other:
3400:Subsets with a maximum (that is, subsets that are
3383:
3272:
3186:
3091:{\displaystyle \{-2^{-n}\,|\,0\leq n<\omega \}}
3090:
3017:
2924:
2886:
2851:
2737:
2684:
2607:
2516:
2412:
2357:
2148:
2038:For an infinite set the order type determines the
1790:
1761:
1735:
1709:
1670:
1611:
1555:
1523:
1496:
1452:
1408:
1367:
1296:
1225:
1169:
1140:
4294:
2637:is not a well ordering, since, for example, the
2202:is an example of well ordering of the integers:
1387:
3577:. Pure and applied mathematics (2nd ed.).
2527:This well order can be visualized as follows:
2053:
3612:
3422:A well-ordered set as topological space is a
2694:does not contain a least element. From the
1652:in the "Antisymmetric" column, respectively.
44:
3411:Subsets that are unbounded in the whole set.
3267:
3261:
3255:
3214:
3181:
3112:
3085:
3044:
3012:
2964:
2846:
2803:
1858:together with the ordering is then called a
2192:integers does not contain a least element.
4270:Positive cone of a partially ordered group
3619:
3605:
3333:Every well-ordered set can be made into a
3295:
3018:{\displaystyle \{1/n\,|\,n=1,2,3,\dots \}}
1905:below for an example). A well-ordered set
51:
37:
3553:
3374:
3239:
3233:
3159:
3153:
3069:
3063:
2984:
2978:
2938:to the natural numbers, which means that
2918:
2877:
2852:{\displaystyle A=\{(x,s(x))\,|\,x\in X\}}
2836:
2830:
2731:
1893:of a well-ordered set, except a possible
4253:Positive cone of an ordered vector space
3530:
3567:
2904:(except possibly for a last element of
2209:one of the following conditions holds:
4295:
3434:), that is, if and only if the set is
2180:Unlike the standard ordering ≤ of the
3600:
2698:axioms of set theory (including the
2043:
1902:
1655:All definitions tacitly require the
1999:Every well-ordered set is uniquely
1946:Every well-ordered set is uniquely
1170:{\displaystyle S\neq \varnothing :}
13:
3780:Properties & Types (
2058:
1988:
14:
4324:
4236:Positive cone of an ordered field
3328:
3314:works for the entire ordered set.
3200:. The previous set is the set of
2184:, the standard ordering ≤ of the
1161:
4090:Ordered topological vector space
3626:
3396:For subsets we can distinguish:
2708:generalized continuum hypothesis
2046:, below, for an example). For a
1640:
1627:
1556:{\displaystyle {\text{not }}aRa}
1061:
1051:
1021:
1011:
989:
979:
944:
922:
912:
877:
845:
835:
810:
778:
763:
743:
711:
701:
691:
671:
639:
619:
609:
599:
567:
547:
503:
483:
473:
436:
411:
401:
369:
344:
307:
277:
245:
183:
148:
2685:{\displaystyle (0,1)\subseteq }
2633:The standard ordering ≤ of any
2467:
2461:
2280:can be visualized as follows:
2063:The standard ordering ≤ of the
16:Class of mathematical orderings
3524:
3489:
3235:
3155:
3065:
2980:
2832:
2827:
2824:
2818:
2806:
2679:
2667:
2661:
2649:
2493:
2485:
2477:
2469:
2457:
2449:
2441:
2433:
2159:This is a well-ordered set of
1648:in the "Symmetric" column and
1584:
1329:
1274:
1203:
1:
4047:Series-parallel partial order
3483:
3384:{\displaystyle \mathbb {N} .}
2887:{\displaystyle \mathbb {Q} .}
1962:, which is equivalent to the
1958:of the well-ordered set. The
1839:with the property that every
1649:
1636:
1046:
1041:
1036:
1031:
1006:
974:
969:
964:
959:
954:
939:
907:
902:
897:
892:
887:
872:
860:
855:
830:
825:
820:
805:
793:
788:
773:
758:
753:
738:
726:
721:
686:
681:
666:
654:
649:
634:
629:
594:
582:
577:
562:
557:
542:
537:
532:
518:
513:
498:
493:
468:
463:
451:
446:
431:
426:
421:
396:
384:
379:
364:
359:
354:
339:
334:
322:
317:
302:
297:
292:
287:
272:
260:
255:
240:
235:
230:
225:
220:
215:
198:
193:
178:
173:
168:
163:
158:
3726:Cantor's isomorphism theorem
2925:{\displaystyle \mathbb {Q} }
2738:{\displaystyle \mathbb {R} }
2054:Examples and counterexamples
7:
3766:Szpilrajn extension theorem
3741:Hausdorff maximal principle
3716:Boolean prime ideal theorem
3498:Information and Computation
3445:
2896:There is an injection from
2413:{\displaystyle x\leq _{z}y}
2175:
10:
4329:
4112:Topological vector lattice
2706:proved that ZF + GCH (the
1992:
1970:), the proof technique of
1909:contains for every subset
1854:in this ordering. The set
4142:
4070:
4009:
3779:
3708:
3657:
3634:
3337:by endowing it with the
3319:axiom of dependent choice
3033:Examples of well orders:
1977:The observation that the
3721:Cantor–Bernstein theorem
3510:10.1016/j.ic.2012.11.003
2628:
4265:Partially ordered group
4085:Specialization preorder
3555:10.4064/fm-56-3-325-345
3541:Fundamenta Mathematicae
3296:Equivalent formulations
2786:is the last element of
2257:are both negative, and
2238:are both positive, and
1985:(for natural numbers).
1983:well-ordering principle
3751:Kruskal's tree theorem
3746:Knaster–Tarski theorem
3736:Dushnik–Miller theorem
3385:
3274:
3188:
3092:
3019:
2926:
2888:
2853:
2739:
2686:
2609:
2518:
2414:
2359:
2150:
2044:§ Natural numbers
1903:§ Natural numbers
1792:
1763:
1737:
1711:
1710:{\displaystyle a,b,c,}
1672:
1613:
1557:
1525:
1498:
1454:
1410:
1369:
1298:
1227:
1171:
1142:
3424:first-countable space
3386:
3312:Transfinite induction
3275:
3189:
3093:
3020:
2927:
2889:
2854:
2740:
2687:
2610:
2519:
2415:
2371:is isomorphic to the
2360:
2151:
1972:transfinite induction
1968:well-founded relation
1960:well-ordering theorem
1793:
1764:
1738:
1712:
1673:
1614:
1558:
1526:
1499:
1455:
1411:
1370:
1299:
1228:
1172:
1143:
1122:Definitions, for all
4243:Ordered vector space
3438:or has the smallest
3370:
3211:
3109:
3041:
2961:
2914:
2873:
2794:
2770:be the successor of
2727:
2646:
2534:
2429:
2391:
2287:
2077:
1791:{\displaystyle aRc.}
1773:
1747:
1721:
1686:
1662:
1657:homogeneous relation
1568:
1536:
1509:
1465:
1421:
1380:
1309:
1238:
1182:
1152:
1126:
868:Strict partial order
143:Equivalence relation
4081:Alexandrov topology
4027:Lexicographic order
3986:Well-quasi-ordering
3581:. pp. 4–6, 9.
3207:The set of numbers
3105:The set of numbers
3037:The set of numbers
2031:-th element" where
1822:well-order relation
1762:{\displaystyle bRc}
1736:{\displaystyle aRb}
1524:{\displaystyle aRa}
1141:{\displaystyle a,b}
527:Well-quasi-ordering
4062:Transitive closure
4022:Converse/Transpose
3731:Dilworth's theorem
3569:Folland, Gerald B.
3468:Well partial order
3381:
3361:, for example the
3270:
3184:
3088:
3015:
2922:
2884:
2861:injective function
2849:
2735:
2682:
2605:
2603:
2514:
2410:
2355:
2353:
2146:
2144:
2048:countably infinite
2023:, the expression "
1941:strict total order
1788:
1759:
1733:
1707:
1668:
1609:
1607:
1553:
1521:
1494:
1492:
1450:
1448:
1406:
1404:
1365:
1363:
1294:
1292:
1223:
1221:
1167:
1138:
1002:Strict total order
4290:
4289:
4248:Partially ordered
4057:Symmetric closure
4042:Reflexive closure
3785:
3588:978-0-471-31716-6
3452:Tree (set theory)
3335:topological space
2704:Wacław Sierpiński
2500:
2465:
2224:is positive, and
1919:least upper bound
1804:
1803:
1671:{\displaystyle R}
1622:
1621:
1594:
1542:
1488:
1444:
1400:
1348:
1257:
935:Strict weak order
121:Total, Semiconnex
4320:
4032:Linear extension
3781:
3761:Mirsky's theorem
3621:
3614:
3607:
3598:
3597:
3592:
3560:
3559:
3557:
3528:
3522:
3521:
3493:
3463:Well-founded set
3454:, generalization
3392:
3390:
3388:
3387:
3382:
3377:
3360:
3286:
3279:
3277:
3276:
3271:
3238:
3232:
3231:
3199:
3193:
3191:
3190:
3185:
3158:
3152:
3151:
3130:
3129:
3101:
3097:
3095:
3094:
3089:
3068:
3062:
3061:
3028:
3024:
3022:
3021:
3016:
2983:
2974:
2953:
2945:
2941:
2937:
2933:
2931:
2929:
2928:
2923:
2921:
2907:
2903:
2899:
2895:
2893:
2891:
2890:
2885:
2880:
2866:
2858:
2856:
2855:
2850:
2835:
2789:
2785:
2781:
2777:
2773:
2769:
2758:
2754:
2750:
2747:well ordered by
2746:
2744:
2742:
2741:
2736:
2734:
2720:
2693:
2691:
2689:
2688:
2683:
2624:
2614:
2612:
2611:
2606:
2604:
2523:
2521:
2520:
2515:
2501:
2498:
2496:
2488:
2480:
2472:
2466:
2463:
2460:
2452:
2444:
2436:
2419:
2417:
2416:
2411:
2406:
2405:
2383:
2370:
2364:
2362:
2361:
2356:
2354:
2279:
2272:
2270:
2264:
2256:
2252:
2247:
2237:
2233:
2227:
2223:
2218:
2205:
2201:
2171:
2155:
2153:
2152:
2147:
2145:
2034:
2030:
2026:
2022:
2001:order isomorphic
1948:order isomorphic
1928:
1924:
1912:
1908:
1900:
1895:greatest element
1892:
1860:well-ordered set
1857:
1849:
1838:
1830:
1797:
1795:
1794:
1789:
1768:
1766:
1765:
1760:
1742:
1740:
1739:
1734:
1716:
1714:
1713:
1708:
1677:
1675:
1674:
1669:
1651:
1647:
1644:
1643:
1638:
1634:
1631:
1630:
1618:
1616:
1615:
1610:
1608:
1595:
1592:
1562:
1560:
1559:
1554:
1543:
1540:
1530:
1528:
1527:
1522:
1503:
1501:
1500:
1495:
1493:
1489:
1486:
1459:
1457:
1456:
1451:
1449:
1445:
1442:
1415:
1413:
1412:
1407:
1405:
1401:
1398:
1374:
1372:
1371:
1366:
1364:
1349:
1346:
1323:
1303:
1301:
1300:
1295:
1293:
1284:
1258:
1255:
1232:
1230:
1229:
1224:
1222:
1207:
1188:
1176:
1174:
1173:
1168:
1147:
1145:
1144:
1139:
1068:
1065:
1064:
1058:
1055:
1054:
1048:
1043:
1038:
1033:
1028:
1025:
1024:
1018:
1015:
1014:
1008:
996:
993:
992:
986:
983:
982:
976:
971:
966:
961:
956:
951:
948:
947:
941:
929:
926:
925:
919:
916:
915:
909:
904:
899:
894:
889:
884:
881:
880:
874:
862:
857:
852:
849:
848:
842:
839:
838:
832:
827:
822:
817:
814:
813:
807:
801:Meet-semilattice
795:
790:
785:
782:
781:
775:
770:
767:
766:
760:
755:
750:
747:
746:
740:
734:Join-semilattice
728:
723:
718:
715:
714:
708:
705:
704:
698:
695:
694:
688:
683:
678:
675:
674:
668:
656:
651:
646:
643:
642:
636:
631:
626:
623:
622:
616:
613:
612:
606:
603:
602:
596:
584:
579:
574:
571:
570:
564:
559:
554:
551:
550:
544:
539:
534:
529:
520:
515:
510:
507:
506:
500:
495:
490:
487:
486:
480:
477:
476:
470:
465:
453:
448:
443:
440:
439:
433:
428:
423:
418:
415:
414:
408:
405:
404:
398:
386:
381:
376:
373:
372:
366:
361:
356:
351:
348:
347:
341:
336:
324:
319:
314:
311:
310:
304:
299:
294:
289:
284:
281:
280:
274:
262:
257:
252:
249:
248:
242:
237:
232:
227:
222:
217:
212:
210:
200:
195:
190:
187:
186:
180:
175:
170:
165:
160:
155:
152:
151:
145:
63:
62:
53:
46:
39:
32:
30:binary relations
21:
20:
4328:
4327:
4323:
4322:
4321:
4319:
4318:
4317:
4313:Wellfoundedness
4308:Ordinal numbers
4293:
4292:
4291:
4286:
4282:Young's lattice
4138:
4066:
4005:
3855:Heyting algebra
3803:Boolean algebra
3775:
3756:Laver's theorem
3704:
3670:Boolean algebra
3665:Binary relation
3653:
3630:
3625:
3595:
3589:
3563:
3529:
3525:
3494:
3490:
3486:
3473:Prewellordering
3448:
3429:
3373:
3371:
3368:
3367:
3365:
3363:natural numbers
3358:
3349:isolated points
3331:
3302:totally ordered
3298:
3281:
3280:has order type
3234:
3224:
3220:
3212:
3209:
3208:
3195:
3194:has order type
3154:
3138:
3134:
3122:
3118:
3110:
3107:
3106:
3099:
3098:has order type
3064:
3054:
3050:
3042:
3039:
3038:
3026:
2979:
2970:
2962:
2959:
2958:
2951:
2946:. For example,
2943:
2939:
2935:
2917:
2915:
2912:
2911:
2909:
2905:
2901:
2897:
2876:
2874:
2871:
2870:
2868:
2864:
2831:
2795:
2792:
2791:
2787:
2783:
2779:
2775:
2771:
2760:
2756:
2752:
2748:
2730:
2728:
2725:
2724:
2722:
2721:is a subset of
2718:
2700:axiom of choice
2647:
2644:
2643:
2641:
2631:
2622:
2602:
2601:
2596:
2591:
2583:
2578:
2570:
2565:
2557:
2552:
2544:
2537:
2535:
2532:
2531:
2499: and
2497:
2492:
2484:
2476:
2468:
2462:
2456:
2448:
2440:
2432:
2430:
2427:
2426:
2401:
2397:
2392:
2389:
2388:
2375:
2368:
2352:
2351:
2346:
2338:
2330:
2322:
2317:
2312:
2307:
2302:
2297:
2290:
2288:
2285:
2284:
2277:
2266:
2265:| ≤ |
2260:
2258:
2254:
2250:
2239:
2235:
2231:
2225:
2221:
2213:
2203:
2199:
2197:binary relation
2182:natural numbers
2178:
2163:
2143:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2080:
2078:
2075:
2074:
2065:natural numbers
2061:
2059:Natural numbers
2056:
2032:
2028:
2024:
2020:
2017:cardinal number
1997:
1991:
1989:Ordinal numbers
1979:natural numbers
1964:axiom of choice
1926:
1922:
1910:
1906:
1898:
1890:
1855:
1847:
1836:
1828:
1806:
1805:
1798:
1774:
1771:
1770:
1748:
1745:
1744:
1722:
1719:
1718:
1687:
1684:
1683:
1663:
1660:
1659:
1653:
1645:
1641:
1632:
1628:
1606:
1605:
1591:
1588:
1587:
1571:
1569:
1566:
1565:
1539:
1537:
1534:
1533:
1510:
1507:
1506:
1491:
1490:
1485:
1482:
1481:
1468:
1466:
1463:
1462:
1447:
1446:
1441:
1438:
1437:
1424:
1422:
1419:
1418:
1403:
1402:
1397:
1394:
1393:
1383:
1381:
1378:
1377:
1362:
1361:
1350:
1345:
1333:
1332:
1324:
1322:
1312:
1310:
1307:
1306:
1291:
1290:
1285:
1283:
1271:
1270:
1259:
1256: and
1254:
1241:
1239:
1236:
1235:
1220:
1219:
1208:
1206:
1200:
1199:
1185:
1183:
1180:
1179:
1153:
1150:
1149:
1127:
1124:
1123:
1066:
1062:
1056:
1052:
1026:
1022:
1016:
1012:
994:
990:
984:
980:
949:
945:
927:
923:
917:
913:
882:
878:
850:
846:
840:
836:
815:
811:
783:
779:
768:
764:
748:
744:
716:
712:
706:
702:
696:
692:
676:
672:
644:
640:
624:
620:
614:
610:
604:
600:
572:
568:
552:
548:
525:
508:
504:
488:
484:
478:
474:
459:Prewellordering
441:
437:
416:
412:
406:
402:
374:
370:
349:
345:
312:
308:
282:
278:
250:
246:
208:
205:
188:
184:
153:
149:
141:
133:
57:
24:
17:
12:
11:
5:
4326:
4316:
4315:
4310:
4305:
4288:
4287:
4285:
4284:
4279:
4274:
4273:
4272:
4262:
4261:
4260:
4255:
4250:
4240:
4239:
4238:
4228:
4223:
4222:
4221:
4216:
4209:Order morphism
4206:
4205:
4204:
4194:
4189:
4184:
4179:
4174:
4173:
4172:
4162:
4157:
4152:
4146:
4144:
4140:
4139:
4137:
4136:
4135:
4134:
4129:
4127:Locally convex
4124:
4119:
4109:
4107:Order topology
4104:
4103:
4102:
4100:Order topology
4097:
4087:
4077:
4075:
4068:
4067:
4065:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4029:
4024:
4019:
4013:
4011:
4007:
4006:
4004:
4003:
3993:
3983:
3978:
3973:
3968:
3963:
3958:
3953:
3948:
3947:
3946:
3936:
3931:
3930:
3929:
3924:
3919:
3914:
3912:Chain-complete
3904:
3899:
3898:
3897:
3892:
3887:
3882:
3877:
3867:
3862:
3857:
3852:
3847:
3837:
3832:
3827:
3822:
3817:
3812:
3811:
3810:
3800:
3795:
3789:
3787:
3777:
3776:
3774:
3773:
3768:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3723:
3718:
3712:
3710:
3706:
3705:
3703:
3702:
3697:
3692:
3687:
3682:
3677:
3672:
3667:
3661:
3659:
3655:
3654:
3652:
3651:
3646:
3641:
3635:
3632:
3631:
3624:
3623:
3616:
3609:
3601:
3594:
3593:
3587:
3564:
3562:
3561:
3548:(3): 325–345.
3523:
3487:
3485:
3482:
3481:
3480:
3475:
3470:
3465:
3460:
3458:Ordinal number
3455:
3447:
3444:
3427:
3413:
3412:
3409:
3405:
3394:
3393:
3380:
3376:
3352:
3339:order topology
3330:
3329:Order topology
3327:
3326:
3325:
3322:
3315:
3309:
3297:
3294:
3293:
3292:
3289:order topology
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3237:
3230:
3227:
3223:
3219:
3216:
3205:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3157:
3150:
3147:
3144:
3141:
3137:
3133:
3128:
3125:
3121:
3117:
3114:
3103:
3087:
3084:
3081:
3078:
3075:
3072:
3067:
3060:
3057:
3053:
3049:
3046:
3031:
3030:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2982:
2977:
2973:
2969:
2966:
2955:
2920:
2883:
2879:
2848:
2845:
2842:
2839:
2834:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2733:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2630:
2627:
2616:
2615:
2600:
2597:
2595:
2592:
2590:
2587:
2584:
2582:
2579:
2577:
2574:
2571:
2569:
2566:
2564:
2561:
2558:
2556:
2553:
2551:
2548:
2545:
2543:
2540:
2539:
2525:
2524:
2513:
2510:
2507:
2504:
2495:
2491:
2487:
2483:
2479:
2475:
2471:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2421:if and only if
2409:
2404:
2400:
2396:
2373:ordinal number
2366:
2365:
2350:
2347:
2345:
2342:
2339:
2337:
2334:
2331:
2329:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2292:
2276:This relation
2274:
2273:
2248:
2229:
2219:
2207:if and only if
2195:The following
2177:
2174:
2157:
2156:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2082:
2060:
2057:
2055:
2052:
2005:ordinal number
1995:Ordinal number
1993:Main article:
1990:
1987:
1952:ordinal number
1833:total ordering
1802:
1801:
1787:
1784:
1781:
1778:
1758:
1755:
1752:
1732:
1729:
1726:
1706:
1703:
1700:
1697:
1694:
1691:
1667:
1624:
1623:
1620:
1619:
1604:
1601:
1598:
1590:
1589:
1586:
1583:
1580:
1577:
1574:
1573:
1563:
1552:
1549:
1546:
1531:
1520:
1517:
1514:
1504:
1484:
1483:
1480:
1477:
1474:
1471:
1470:
1460:
1440:
1439:
1436:
1433:
1430:
1427:
1426:
1416:
1396:
1395:
1392:
1389:
1386:
1385:
1375:
1360:
1357:
1354:
1351:
1347: or
1344:
1341:
1338:
1335:
1334:
1331:
1328:
1325:
1321:
1318:
1315:
1314:
1304:
1289:
1286:
1282:
1279:
1276:
1273:
1272:
1269:
1266:
1263:
1260:
1253:
1250:
1247:
1244:
1243:
1233:
1218:
1215:
1212:
1209:
1205:
1202:
1201:
1198:
1195:
1192:
1189:
1187:
1177:
1166:
1163:
1160:
1157:
1137:
1134:
1131:
1119:
1118:
1113:
1108:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1070:
1069:
1059:
1049:
1044:
1039:
1034:
1029:
1019:
1009:
1004:
998:
997:
987:
977:
972:
967:
962:
957:
952:
942:
937:
931:
930:
920:
910:
905:
900:
895:
890:
885:
875:
870:
864:
863:
858:
853:
843:
833:
828:
823:
818:
808:
803:
797:
796:
791:
786:
776:
771:
761:
756:
751:
741:
736:
730:
729:
724:
719:
709:
699:
689:
684:
679:
669:
664:
658:
657:
652:
647:
637:
632:
627:
617:
607:
597:
592:
586:
585:
580:
575:
565:
560:
555:
545:
540:
535:
530:
522:
521:
516:
511:
501:
496:
491:
481:
471:
466:
461:
455:
454:
449:
444:
434:
429:
424:
419:
409:
399:
394:
388:
387:
382:
377:
367:
362:
357:
352:
342:
337:
332:
330:Total preorder
326:
325:
320:
315:
305:
300:
295:
290:
285:
275:
270:
264:
263:
258:
253:
243:
238:
233:
228:
223:
218:
213:
202:
201:
196:
191:
181:
176:
171:
166:
161:
156:
146:
138:
137:
135:
130:
128:
126:
124:
122:
119:
117:
115:
112:
111:
106:
101:
96:
91:
86:
81:
76:
71:
66:
59:
58:
56:
55:
48:
41:
33:
19:
18:
15:
9:
6:
4:
3:
2:
4325:
4314:
4311:
4309:
4306:
4304:
4301:
4300:
4298:
4283:
4280:
4278:
4275:
4271:
4268:
4267:
4266:
4263:
4259:
4256:
4254:
4251:
4249:
4246:
4245:
4244:
4241:
4237:
4234:
4233:
4232:
4231:Ordered field
4229:
4227:
4224:
4220:
4217:
4215:
4212:
4211:
4210:
4207:
4203:
4200:
4199:
4198:
4195:
4193:
4190:
4188:
4187:Hasse diagram
4185:
4183:
4180:
4178:
4175:
4171:
4168:
4167:
4166:
4165:Comparability
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4147:
4145:
4141:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4114:
4113:
4110:
4108:
4105:
4101:
4098:
4096:
4093:
4092:
4091:
4088:
4086:
4082:
4079:
4078:
4076:
4073:
4069:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4037:Product order
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4014:
4012:
4010:Constructions
4008:
4002:
3998:
3994:
3991:
3987:
3984:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3945:
3942:
3941:
3940:
3937:
3935:
3932:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3909:
3908:
3907:Partial order
3905:
3903:
3900:
3896:
3895:Join and meet
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3872:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3842:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3809:
3806:
3805:
3804:
3801:
3799:
3796:
3794:
3793:Antisymmetric
3791:
3790:
3788:
3784:
3778:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3713:
3711:
3707:
3701:
3700:Weak ordering
3698:
3696:
3693:
3691:
3688:
3686:
3685:Partial order
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3660:
3656:
3650:
3647:
3645:
3642:
3640:
3637:
3636:
3633:
3629:
3622:
3617:
3615:
3610:
3608:
3603:
3602:
3599:
3590:
3584:
3580:
3576:
3575:
3570:
3566:
3565:
3556:
3551:
3547:
3543:
3542:
3537:
3533:
3527:
3519:
3515:
3511:
3507:
3503:
3499:
3492:
3488:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3453:
3450:
3449:
3443:
3441:
3437:
3433:
3425:
3420:
3418:
3410:
3406:
3403:
3399:
3398:
3397:
3378:
3364:
3356:
3353:
3350:
3347:
3346:
3345:
3342:
3340:
3336:
3323:
3320:
3316:
3313:
3310:
3307:
3306:
3305:
3303:
3290:
3284:
3264:
3258:
3252:
3249:
3246:
3243:
3240:
3228:
3225:
3221:
3217:
3206:
3203:
3198:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3148:
3145:
3142:
3139:
3135:
3131:
3126:
3123:
3119:
3115:
3104:
3082:
3079:
3076:
3073:
3070:
3058:
3055:
3051:
3047:
3036:
3035:
3034:
3009:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2975:
2971:
2967:
2956:
2949:
2948:
2947:
2881:
2862:
2843:
2840:
2837:
2821:
2815:
2812:
2809:
2800:
2797:
2767:
2763:
2715:
2713:
2709:
2705:
2701:
2697:
2676:
2673:
2670:
2664:
2658:
2655:
2652:
2640:
2639:open interval
2636:
2635:real interval
2626:
2621:
2618:This has the
2598:
2593:
2588:
2585:
2580:
2575:
2572:
2567:
2562:
2559:
2554:
2549:
2546:
2541:
2530:
2529:
2528:
2511:
2508:
2505:
2502:
2489:
2481:
2473:
2453:
2445:
2437:
2425:
2424:
2423:
2422:
2407:
2402:
2398:
2394:
2385:
2382:
2378:
2374:
2348:
2343:
2340:
2335:
2332:
2327:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2283:
2282:
2281:
2269:
2263:
2249:
2246:
2242:
2230:
2220:
2216:
2212:
2211:
2210:
2208:
2198:
2193:
2191:
2187:
2183:
2173:
2170:
2166:
2162:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2073:
2072:
2071:
2068:
2066:
2051:
2049:
2045:
2041:
2036:
2018:
2014:
2010:
2007:, called the
2006:
2002:
1996:
1986:
1984:
1980:
1975:
1973:
1969:
1965:
1961:
1957:
1954:, called the
1953:
1949:
1944:
1942:
1939:
1935:
1930:
1920:
1916:
1904:
1896:
1887:
1885:
1884:well ordering
1881:
1877:
1873:
1869:
1865:
1861:
1853:
1852:least element
1845:
1842:
1834:
1827:
1823:
1819:
1818:well-ordering
1815:
1811:
1800:
1785:
1782:
1779:
1776:
1756:
1753:
1750:
1730:
1727:
1724:
1704:
1701:
1698:
1695:
1692:
1689:
1681:
1665:
1658:
1626:
1625:
1602:
1599:
1596:
1581:
1578:
1575:
1564:
1550:
1547:
1544:
1532:
1518:
1515:
1512:
1505:
1478:
1475:
1472:
1461:
1434:
1431:
1428:
1417:
1390:
1376:
1358:
1355:
1352:
1342:
1339:
1336:
1326:
1319:
1316:
1305:
1287:
1280:
1277:
1267:
1264:
1261:
1251:
1248:
1245:
1234:
1216:
1213:
1210:
1196:
1193:
1190:
1178:
1164:
1158:
1155:
1135:
1132:
1129:
1121:
1120:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1081:Antisymmetric
1079:
1077:
1074:
1072:
1071:
1060:
1050:
1045:
1040:
1035:
1030:
1020:
1010:
1005:
1003:
1000:
999:
988:
978:
973:
968:
963:
958:
953:
943:
938:
936:
933:
932:
921:
911:
906:
901:
896:
891:
886:
876:
871:
869:
866:
865:
859:
854:
844:
834:
829:
824:
819:
809:
804:
802:
799:
798:
792:
787:
777:
772:
762:
757:
752:
742:
737:
735:
732:
731:
725:
720:
710:
700:
690:
685:
680:
670:
665:
663:
660:
659:
653:
648:
638:
633:
628:
618:
608:
598:
593:
591:
590:Well-ordering
588:
587:
581:
576:
566:
561:
556:
546:
541:
536:
531:
528:
524:
523:
517:
512:
502:
497:
492:
482:
472:
467:
462:
460:
457:
456:
450:
445:
435:
430:
425:
420:
410:
400:
395:
393:
390:
389:
383:
378:
368:
363:
358:
353:
343:
338:
333:
331:
328:
327:
321:
316:
306:
301:
296:
291:
286:
276:
271:
269:
268:Partial order
266:
265:
259:
254:
244:
239:
234:
229:
224:
219:
214:
211:
204:
203:
197:
192:
182:
177:
172:
167:
162:
157:
147:
144:
140:
139:
136:
131:
129:
127:
125:
123:
120:
118:
116:
114:
113:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
75:
74:Antisymmetric
72:
70:
67:
65:
64:
61:
60:
54:
49:
47:
42:
40:
35:
34:
31:
27:
23:
22:
4303:Order theory
4074:& Orders
4052:Star product
4000:
3981:Well-founded
3934:Prefix order
3890:Distributive
3880:Complemented
3850:Foundational
3815:Completeness
3771:Zorn's lemma
3675:Cyclic order
3658:Key concepts
3628:Order theory
3573:
3545:
3539:
3532:Feferman, S.
3526:
3501:
3497:
3491:
3478:Directed set
3442:order type.
3421:
3415:A subset is
3414:
3395:
3355:limit points
3343:
3332:
3300:If a set is
3299:
3282:
3202:limit points
3196:
3032:
2778:ordering on
2765:
2761:
2716:
2632:
2617:
2526:
2386:
2380:
2376:
2367:
2275:
2267:
2261:
2244:
2240:
2214:
2194:
2179:
2168:
2164:
2158:
2069:
2062:
2037:
2003:to a unique
1998:
1976:
1950:to a unique
1945:
1938:well-founded
1931:
1888:
1883:
1880:well ordered
1879:
1875:
1872:wellordering
1871:
1867:
1863:
1859:
1821:
1817:
1813:
1807:
1654:
1091:Well-founded
589:
209:(Quasiorder)
84:Well-founded
4258:Riesz space
4219:Isomorphism
4095:Normal cone
4017:Composition
3951:Semilattice
3860:Homogeneous
3845:Equivalence
3695:Total order
3440:uncountable
3287:. With the
2751:. For each
2228:is negative
2040:cardinality
1915:upper bound
1868:wellordered
1810:mathematics
1111:Irreflexive
392:Total order
104:Irreflexive
4297:Categories
4226:Order type
4160:Cofinality
4001:Well-order
3976:Transitive
3865:Idempotent
3798:Asymmetric
3484:References
2620:order type
2161:order type
2009:order type
1956:order type
1934:non-strict
1932:If ≤ is a
1876:well order
1814:well-order
1682:: for all
1680:transitive
1116:Asymmetric
109:Asymmetric
26:Transitive
4277:Upper set
4214:Embedding
4150:Antichain
3971:Tolerance
3961:Symmetric
3956:Semiorder
3902:Reflexive
3820:Connected
3436:countable
3432:omega-one
3259:∪
3253:ω
3244:≤
3226:−
3218:−
3179:ω
3164:≤
3146:−
3140:−
3132:−
3124:−
3116:−
3083:ω
3074:≤
3056:−
3048:−
3010:…
2841:∈
2665:⊆
2599:…
2586:−
2573:−
2560:−
2547:−
2506:≤
2399:≤
2349:…
2341:−
2333:−
2325:−
2320:…
2140:…
2110:…
1864:wellorder
1841:non-empty
1593:not
1585:⇒
1541:not
1476:∧
1432:∨
1330:⇒
1320:≠
1275:⇒
1204:⇒
1162:∅
1159:≠
1106:Reflexive
1101:Has meets
1096:Has joins
1086:Connected
1076:Symmetric
207:Preorder
134:reflexive
99:Reflexive
94:Has meets
89:Has joins
79:Connected
69:Symmetric
4072:Topology
3939:Preorder
3922:Eulerian
3885:Complete
3835:Directed
3825:Covering
3690:Preorder
3649:Category
3644:Glossary
3571:(1999).
3534:(1964).
3504:: 1–22.
3446:See also
2957:The set
2782:(unless
2190:negative
2186:integers
2176:Integers
2013:counting
1913:with an
1650:✗
1637:✗
1047:✗
1042:✗
1037:✗
1032:✗
1007:✗
975:✗
970:✗
965:✗
960:✗
955:✗
940:✗
908:✗
903:✗
898:✗
893:✗
888:✗
873:✗
861:✗
856:✗
831:✗
826:✗
821:✗
806:✗
794:✗
789:✗
774:✗
759:✗
754:✗
739:✗
727:✗
722:✗
687:✗
682:✗
667:✗
655:✗
650:✗
635:✗
630:✗
595:✗
583:✗
578:✗
563:✗
558:✗
543:✗
538:✗
533:✗
519:✗
514:✗
499:✗
494:✗
469:✗
464:✗
452:✗
447:✗
432:✗
427:✗
422:✗
397:✗
385:✗
380:✗
365:✗
360:✗
355:✗
340:✗
335:✗
323:✗
318:✗
303:✗
298:✗
293:✗
288:✗
273:✗
261:✗
256:✗
241:✗
236:✗
231:✗
226:✗
221:✗
216:✗
199:✗
194:✗
179:✗
174:✗
169:✗
164:✗
159:✗
4177:Duality
4155:Cofinal
4143:Related
4122:Fréchet
3999:)
3875:Bounded
3870:Lattice
3843:)
3841:Partial
3709:Results
3680:Lattice
3518:3016456
3417:cofinal
3402:bounded
3391:
3366:
2932:
2910:
2894:
2869:
2790:). Let
2745:
2723:
2692:
2642:
1824:) on a
662:Lattice
4202:Subnet
4182:Filter
4132:Normed
4117:Banach
4083:&
3990:Better
3927:Strict
3917:Graded
3808:topics
3639:Topics
3585:
3516:
2759:, let
2271:|
2259:|
1882:, and
1870:, and
1850:has a
1844:subset
1487:exists
1443:exists
1399:exists
28:
4192:Ideal
4170:Graph
3966:Total
3944:Total
3830:Dense
3579:Wiley
2863:from
2629:Reals
2204:x R y
1831:is a
1769:then
132:Anti-
3783:list
3583:ISBN
3408:set.
3250:<
3176:<
3080:<
2446:<
2253:and
2234:and
1816:(or
1812:, a
1743:and
1148:and
4197:Net
3997:Pre
3550:doi
3506:doi
3502:224
3285:+ 1
2900:to
2867:to
2774:in
2755:in
2712:V=L
2696:ZFC
2217:= 0
1925:in
1874:or
1846:of
1835:on
1826:set
1820:or
1808:In
1717:if
1678:be
1388:min
4299::
3546:56
3544:.
3538:.
3514:MR
3512:.
3500:.
3341:.
3321:).
2625:.
2464:or
2384:.
2379:+
2243:≤
2167:+
1929:.
1917:a
1886:.
1878:,
1866:,
3995:(
3992:)
3988:(
3839:(
3786:)
3620:e
3613:t
3606:v
3591:.
3558:.
3552::
3520:.
3508::
3430:(
3428:1
3379:.
3375:N
3359:ω
3283:ω
3268:}
3265:1
3262:{
3256:}
3247:n
3241:0
3236:|
3229:n
3222:2
3215:{
3197:ω
3182:}
3173:n
3170:,
3167:m
3161:0
3156:|
3149:n
3143:m
3136:2
3127:n
3120:2
3113:{
3102:.
3100:ω
3086:}
3077:n
3071:0
3066:|
3059:n
3052:2
3045:{
3029:.
3027:≤
3013:}
3007:,
3004:3
3001:,
2998:2
2995:,
2992:1
2989:=
2986:n
2981:|
2976:n
2972:/
2968:1
2965:{
2954:.
2952:≤
2944:≤
2940:X
2936:X
2919:Q
2906:X
2902:A
2898:X
2882:.
2878:Q
2865:A
2847:}
2844:X
2838:x
2833:|
2828:)
2825:)
2822:x
2819:(
2816:s
2813:,
2810:x
2807:(
2804:{
2801:=
2798:A
2788:X
2784:x
2780:X
2776:≤
2772:x
2768:)
2766:x
2764:(
2762:s
2757:X
2753:x
2749:≤
2732:R
2719:X
2680:]
2677:1
2674:,
2671:0
2668:[
2662:)
2659:1
2656:,
2653:0
2650:(
2623:ω
2594:4
2589:4
2581:3
2576:3
2568:2
2563:2
2555:1
2550:1
2542:0
2512:.
2509:y
2503:x
2494:|
2490:y
2486:|
2482:=
2478:|
2474:x
2470:|
2458:|
2454:y
2450:|
2442:|
2438:x
2434:|
2408:y
2403:z
2395:x
2381:ω
2377:ω
2369:R
2344:3
2336:2
2328:1
2315:4
2310:3
2305:2
2300:1
2295:0
2278:R
2268:y
2262:x
2255:y
2251:x
2245:y
2241:x
2236:y
2232:x
2226:y
2222:x
2215:x
2200:R
2169:ω
2165:ω
2135:9
2130:7
2125:5
2120:3
2115:1
2105:8
2100:6
2095:4
2090:2
2085:0
2033:β
2029:β
2025:n
2021:n
1927:S
1923:T
1911:T
1907:S
1899:s
1891:s
1856:S
1848:S
1837:S
1829:S
1786:.
1783:c
1780:R
1777:a
1757:c
1754:R
1751:b
1731:b
1728:R
1725:a
1705:,
1702:c
1699:,
1696:b
1693:,
1690:a
1666:R
1646:Y
1633:Y
1603:a
1600:R
1597:b
1582:b
1579:R
1576:a
1551:a
1548:R
1545:a
1519:a
1516:R
1513:a
1479:b
1473:a
1435:b
1429:a
1391:S
1359:a
1356:R
1353:b
1343:b
1340:R
1337:a
1327:b
1317:a
1288:b
1281:=
1278:a
1268:a
1265:R
1262:b
1252:b
1249:R
1246:a
1217:a
1214:R
1211:b
1197:b
1194:R
1191:a
1165::
1156:S
1136:b
1133:,
1130:a
1067:Y
1057:Y
1027:Y
1017:Y
995:Y
985:Y
950:Y
928:Y
918:Y
883:Y
851:Y
841:Y
816:Y
784:Y
769:Y
749:Y
717:Y
707:Y
697:Y
677:Y
645:Y
625:Y
615:Y
605:Y
573:Y
553:Y
509:Y
489:Y
479:Y
442:Y
417:Y
407:Y
375:Y
350:Y
313:Y
283:Y
251:Y
189:Y
154:Y
52:e
45:t
38:v
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