33:
2128:
5110:
5090:, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special
6727:
175:
5931:
5601:
5501:
5888:
6467:
5469:
5441:
4516:
2255:
2179:
991:
382:
5145:
6612:
6503:
5971:
5569:
5218:
4926:
1968:
6559:
6427:
6371:
4387:
4090:
912:
7047:
5174:
5019:
4858:
4483:
4439:
4288:
4252:
4168:
4122:
4031:
3952:
3880:
3733:
3607:
3378:
3212:
3123:
3064:
2977:
2746:
2601:
2432:
2222:
1359:
1049:
958:
863:
732:
349:
7009:
6948:
6890:
6082:
5331:
5303:
5275:
5247:
3324:
3006:
2778:
2567:
2397:
2368:
32:
6861:
6193:
6138:
6112:
6053:
5785:
5363:
4214:
3799:
3551:
3525:
3479:
3424:
3275:
3245:
2868:
2834:
2808:
2538:
1729:
1700:
1390:
1152:
1123:
697:
668:
518:
489:
6835:
6809:
6662:
5640:
3842:
3701:
3180:
3032:
2945:
2512:
2462:
2339:
2281:
1674:
1247:
1097:
639:
587:
460:
408:
6754:
5848:
5821:
5537:
2631:
6919:
3453:
1566:
6239:
6526:
6394:
6318:
5739:
5406:
5042:
4997:
4738:
4613:
3999:
3150:
3091:
1537:
1219:
310:
6968:
6774:
6632:
6579:
6338:
6295:
6259:
6213:
6167:
6024:
6004:
5759:
5716:
5383:
4974:
4949:
4878:
4829:
4809:
4785:
4760:
4715:
4695:
4675:
4655:
4635:
4590:
4566:
4539:
4407:
4352:
4328:
4308:
4188:
4146:
4051:
3972:
3920:
3900:
3819:
3773:
3753:
3671:
3651:
3627:
3571:
3499:
3398:
3352:
3295:
2908:
2888:
2714:
2694:
2482:
2307:
2105:
2084:
2056:
2036:
2012:
1988:
1941:
1920:
1899:
1878:
1858:
1834:
1813:
1793:
1773:
1753:
1648:
1627:
1606:
1586:
1510:
1490:
1470:
1450:
1430:
1410:
1327:
1307:
1287:
1267:
1192:
1172:
1071:
1017:
827:
801:
776:
756:
611:
561:
541:
432:
287:
259:
239:
215:
101:
81:
57:
2635:
A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.
5652:
7097:
2181:
has two maximal elements, viz. 3 and 4, none of which is greatest. It has one minimal element, viz. 1, which is also its least element.
7083:
5667:
6667:
2127:
5662:
5657:
178:
3152:
This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important
5091:
5415:, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
7089:
2910:
is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are
2111:
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative
4788:
758:
can have at most one greatest element and it can have at most one least element. Whenever a greatest element of
5471:
the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
106:
5673:
5066:
The least and greatest element of the whole partially ordered set play a special role and are also called
4928:
in the topmost picture is an example), then the notions of maximal element and greatest element coincide.
996:
266:
7121:
7116:
5604:
4486:
2435:
735:
218:
5893:
5577:
5477:
5853:
5087:
5054:
6432:
5449:
5421:
4492:
2231:
2134:
967:
358:
5128:
6584:
6472:
5936:
5542:
5179:
4887:
2841:
2189:
of the set, which are elements that are not strictly smaller than any other element in the set.
1947:
6531:
6399:
6343:
4357:
4060:
882:
7014:
5157:
5002:
4841:
4456:
4412:
4261:
4222:
4151:
4095:
4004:
3925:
3853:
3706:
3580:
3361:
3185:
3096:
3037:
2950:
2719:
2574:
2405:
2195:
1332:
1022:
931:
836:
705:
322:
7093:
7070:
6982:
6924:
6866:
6058:
5307:
5279:
5251:
5223:
3300:
2982:
2751:
2543:
2373:
2344:
2116:
6840:
6172:
6117:
6091:
6029:
5764:
5336:
4193:
3778:
3530:
3504:
3458:
3403:
3254:
3221:
2847:
2813:
2787:
2517:
1705:
1679:
1366:
1128:
1102:
673:
647:
494:
468:
6814:
6788:
6637:
5610:
4956:
If the notions of maximal element and greatest element coincide on every two-element subset
3824:
3680:
3159:
3011:
2924:
2491:
2441:
2318:
2260:
1653:
1226:
1076:
618:
566:
439:
387:
6732:
5826:
5799:
5510:
2607:
6895:
5412:
4832:
3429:
2286:
2186:
1542:
177:
has one greatest element, viz. 30, and one least element, viz. 1. These elements are also
6218:
6508:
6376:
6300:
5721:
5443:
the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
5388:
5024:
4979:
4720:
4595:
3981:
3132:
3073:
1519:
1201:
292:
6953:
6759:
6617:
6564:
6323:
6280:
6244:
6198:
6152:
6009:
5989:
5744:
5701:
5368:
4959:
4934:
4863:
4814:
4794:
4770:
4745:
4700:
4680:
4660:
4640:
4620:
4575:
4551:
4524:
4392:
4337:
4313:
4293:
4173:
4131:
4036:
3957:
3905:
3885:
3848:
3804:
3758:
3738:
3656:
3636:
3612:
3556:
3484:
3383:
3337:
3280:
2893:
2873:
2699:
2679:
2467:
2292:
2225:
2119:(the number 0 in this case) does not imply the existence of a greatest element either.
2090:
2069:
2041:
2021:
1997:
1973:
1926:
1905:
1884:
1863:
1843:
1819:
1798:
1778:
1758:
1738:
1633:
1612:
1591:
1571:
1495:
1475:
1455:
1435:
1415:
1395:
1312:
1292:
1272:
1252:
1177:
1157:
1056:
1002:
961:
812:
786:
761:
741:
596:
546:
526:
417:
352:
272:
244:
224:
200:
86:
66:
42:
7110:
5504:
5113:
4545:
greatest element. Thus if a set has a greatest element then it is necessarily unique.
2664:
2651:
36:
6085:
5974:
5098:
4255:
2915:
190:
4092:
However, the uniqueness conclusion is no longer guaranteed if the preordered set
2185:
A greatest element of a subset of a preordered set should not be confused with a
5148:
4881:
4569:
2639:
2112:
922:
865:
has a greatest element (resp. a least element) then this element is also called
186:
7049:
would have two maximal, but no greatest element, contradicting the coincidence.
5678:
2642:
the maximal element and the greatest element coincide; and it is also called
17:
5681:— a non-strict order such that every non-empty set has a least element
3553:
holds, which shows that all pairs of distinct (i.e. non-equal) elements in
2672:
Role of (in)comparability in distinguishing greatest vs. maximal elements
5698:
Of course, in this particular example, there exists only one element in
5109:
3609:
can not possibly have a greatest element (because a greatest element of
7069:
The notion of locality requires the function's domain to be at least a
5122:
60:
6892:
In the second case, the definition of maximal element requires that
4762:
has several maximal elements then it cannot have a greatest element.
543:
is on in the above definition, the definition of a least element of
4951:
has a greatest element, the notions coincide, too, as stated above.
2748:
has to do with what elements they are comparable to. Two elements
5108:
2918:
partially ordered sets) may have elements that are incomparable.
2126:
31:
6776:
and not maximal). This contradicts the ascending chain condition.
5086:. The notation of 0 and 1 is used preferably when the poset is a
2676:
One of the most important differences between a greatest element
5408:
but no least upper bound, and no greatest element (cf. picture).
4033:
is also partially ordered then it is possible to conclude that
3093:
This is not required of maximal elements. Maximal elements of
6722:{\displaystyle s_{1}<s_{2}<\cdots <s_{n}<\cdots }
5097:
Further introductory information is found in the article on
3330:
Example where all elements are maximal but none are greatest
1735:
to the definition of a greatest element given before. Thus
4354:
is a greatest element (and thus also a maximal element) of
2115:. This example also demonstrates that the existence of a
5082:(1), respectively. If both exist, the poset is called a
4931:
However, this is not a necessary condition for whenever
2914:
pairs. In general, however, preordered sets (and even
2123:
Contrast to maximal elements and local/absolute maximums
3954:
and moreover, as a consequence of the greatest element
2646:; in the case of function values it is also called the
7017:
6985:
6956:
6927:
6898:
6869:
6843:
6817:
6791:
6762:
6735:
6670:
6664:
Repeating this argument, an infinite ascending chain
6640:
6620:
6587:
6567:
6534:
6511:
6475:
6435:
6402:
6379:
6346:
6326:
6303:
6283:
6247:
6221:
6201:
6175:
6155:
6120:
6094:
6061:
6032:
6012:
5992:
5939:
5896:
5856:
5829:
5802:
5767:
5747:
5724:
5704:
5613:
5580:
5545:
5513:
5480:
5452:
5424:
5391:
5371:
5339:
5310:
5282:
5254:
5226:
5182:
5160:
5131:
5027:
5005:
4982:
4962:
4937:
4890:
4866:
4844:
4817:
4797:
4773:
4748:
4723:
4703:
4683:
4663:
4643:
4623:
4598:
4578:
4554:
4527:
4495:
4459:
4415:
4395:
4360:
4340:
4316:
4296:
4264:
4225:
4196:
4176:
4154:
4134:
4098:
4063:
4039:
4007:
3984:
3960:
3928:
3908:
3888:
3856:
3827:
3807:
3781:
3761:
3741:
3709:
3683:
3659:
3639:
3615:
3583:
3559:
3533:
3507:
3487:
3461:
3432:
3406:
3386:
3364:
3340:
3303:
3283:
3257:
3224:
3188:
3162:
3135:
3099:
3076:
3040:
3014:
2985:
2953:
2927:
2896:
2876:
2850:
2816:
2790:
2754:
2722:
2702:
2682:
2610:
2577:
2546:
2520:
2494:
2470:
2444:
2408:
2376:
2347:
2321:
2295:
2263:
2234:
2198:
2137:
2093:
2072:
2044:
2024:
2014:(however, it may be possible that some other element
2000:
1976:
1950:
1929:
1908:
1887:
1866:
1846:
1822:
1801:
1781:
1761:
1741:
1708:
1682:
1656:
1636:
1615:
1594:
1574:
1545:
1522:
1498:
1478:
1458:
1438:
1418:
1398:
1369:
1335:
1315:
1295:
1275:
1255:
1229:
1204:
1180:
1160:
1131:
1105:
1079:
1059:
1025:
1005:
970:
934:
885:
839:
815:
789:
764:
744:
708:
676:
650:
621:
599:
569:
549:
529:
497:
471:
442:
420:
390:
361:
325:
295:
275:
247:
227:
203:
109:
89:
69:
45:
4310:
has exactly one element. All pairs of elements from
4128:also partially ordered. For example, suppose that
3034:; so by its very definition, a greatest element of
2840:if they are not comparable. Because preorders are
2604:is defined to mean a maximal element of the subset
7041:
7003:
6962:
6942:
6913:
6884:
6855:
6829:
6803:
6768:
6748:
6721:
6656:
6626:
6606:
6573:
6553:
6520:
6497:
6461:
6421:
6388:
6365:
6332:
6312:
6289:
6253:
6233:
6207:
6187:
6161:
6132:
6106:
6076:
6047:
6018:
5998:
5965:
5925:
5882:
5842:
5815:
5779:
5753:
5733:
5710:
5634:
5595:
5563:
5531:
5495:
5463:
5435:
5400:
5377:
5357:
5325:
5297:
5269:
5241:
5212:
5168:
5139:
5036:
5013:
4991:
4968:
4943:
4920:
4872:
4852:
4823:
4803:
4779:
4754:
4732:
4709:
4689:
4669:
4649:
4629:
4607:
4584:
4560:
4533:
4510:
4477:
4433:
4401:
4381:
4346:
4322:
4302:
4282:
4246:
4208:
4182:
4162:
4140:
4116:
4084:
4045:
4025:
3993:
3966:
3946:
3914:
3894:
3874:
3836:
3813:
3793:
3767:
3747:
3727:
3695:
3665:
3645:
3621:
3601:
3565:
3545:
3519:
3493:
3473:
3447:
3418:
3392:
3372:
3346:
3318:
3289:
3269:
3239:
3206:
3174:
3144:
3117:
3085:
3058:
3026:
3000:
2971:
2939:
2902:
2882:
2862:
2828:
2802:
2772:
2740:
2708:
2688:
2625:
2595:
2561:
2532:
2506:
2476:
2456:
2426:
2391:
2362:
2333:
2301:
2275:
2249:
2216:
2173:
2099:
2078:
2050:
2030:
2006:
1982:
1962:
1935:
1914:
1893:
1872:
1852:
1828:
1807:
1787:
1767:
1747:
1723:
1694:
1668:
1642:
1621:
1600:
1580:
1560:
1531:
1504:
1484:
1464:
1444:
1424:
1404:
1384:
1353:
1321:
1301:
1281:
1261:
1241:
1213:
1186:
1166:
1146:
1117:
1091:
1065:
1043:
1011:
985:
952:
906:
857:
821:
795:
770:
750:
726:
691:
662:
633:
605:
581:
555:
535:
512:
483:
454:
426:
402:
376:
343:
304:
281:
253:
233:
209:
169:
95:
75:
51:
778:exists and is unique then this element is called
2668:. Similar conclusions hold for least elements.
2131:In the above divisibility order, the red subset
3629:would, in particular, have to be comparable to
3358:(distinct) elements and define a partial order
3129:required to be comparable to every element in
8:
7036:
7024:
5352:
5340:
5207:
5183:
4915:
4897:
2168:
2144:
289:that is smaller than every other element of
241:that is greater than every other element of
164:
116:
4697:and moreover, any other maximal element of
4548:If it exists, then the greatest element of
523:By switching the side of the relation that
5057:always has a greatest and a least element.
63:of 60, partially ordered by the relation "
7016:
6984:
6955:
6926:
6897:
6868:
6842:
6816:
6790:
6761:
6740:
6734:
6707:
6688:
6675:
6669:
6645:
6639:
6619:
6592:
6586:
6566:
6545:
6533:
6510:
6480:
6474:
6453:
6440:
6434:
6407:
6401:
6378:
6351:
6345:
6325:
6302:
6282:
6246:
6220:
6200:
6174:
6154:
6119:
6093:
6060:
6031:
6011:
5991:
5957:
5944:
5938:
5914:
5901:
5895:
5874:
5861:
5855:
5834:
5828:
5807:
5801:
5766:
5746:
5723:
5703:
5612:
5587:
5583:
5582:
5579:
5544:
5512:
5487:
5483:
5482:
5479:
5454:
5453:
5451:
5426:
5425:
5423:
5390:
5370:
5338:
5309:
5281:
5253:
5225:
5181:
5165:
5161:
5159:
5133:
5132:
5130:
5026:
5010:
5006:
5004:
4981:
4961:
4936:
4889:
4865:
4849:
4845:
4843:
4816:
4796:
4772:
4747:
4722:
4702:
4682:
4662:
4642:
4622:
4597:
4577:
4553:
4526:
4494:
4458:
4414:
4394:
4359:
4339:
4315:
4295:
4263:
4224:
4195:
4175:
4159:
4155:
4153:
4148:is a non-empty set and define a preorder
4133:
4097:
4062:
4038:
4006:
3983:
3959:
3927:
3922:will necessarily be a maximal element of
3907:
3887:
3855:
3826:
3806:
3780:
3760:
3740:
3708:
3682:
3658:
3638:
3614:
3582:
3558:
3532:
3506:
3486:
3460:
3431:
3405:
3385:
3369:
3365:
3363:
3339:
3302:
3282:
3256:
3223:
3187:
3161:
3134:
3098:
3075:
3039:
3013:
2984:
2952:
2926:
2895:
2875:
2849:
2815:
2789:
2753:
2721:
2701:
2681:
2609:
2576:
2545:
2519:
2493:
2469:
2443:
2407:
2375:
2346:
2320:
2311:if the following condition is satisfied:
2294:
2262:
2233:
2197:
2136:
2092:
2071:
2043:
2023:
1999:
1975:
1949:
1928:
1907:
1886:
1865:
1845:
1821:
1800:
1780:
1760:
1740:
1707:
1681:
1655:
1635:
1614:
1593:
1573:
1544:
1521:
1497:
1477:
1457:
1437:
1417:
1397:
1368:
1334:
1314:
1294:
1274:
1254:
1228:
1203:
1179:
1159:
1130:
1104:
1078:
1058:
1024:
1004:
969:
933:
921:Greatest elements are closely related to
884:
838:
814:
788:
763:
743:
707:
675:
649:
620:
598:
568:
548:
528:
496:
470:
441:
419:
389:
360:
324:
294:
274:
246:
226:
202:
108:
88:
68:
44:
5653:Essential supremum and essential infimum
3735:because there is exactly one element in
170:{\displaystyle S=\{1,2,3,5,6,10,15,30\}}
7082:Davey, B. A.; Priestley, H. A. (2002).
7062:
5691:
3882:does happen to have a greatest element
3156:statement. The defining condition for
2066:for there to exist some upper bound of
1392:In particular, any greatest element of
3277:(so elements that are incomparable to
3066:must, in particular, be comparable to
2038:). In particular, it is possible for
5761:itself, so the second condition "and
7:
4290:is partially ordered if and only if
563:is obtained. Explicitly, an element
6614:contradicts the incomparability of
6505:The latter must be incomparable to
6373:must exist that is incomparable to
29:Element ≥ (or ≤) each other element
7085:Introduction to Lattices and Order
5607:, this set has upper bounds, e.g.
917:Relationship to upper/lower bounds
181:, respectively, of the red subset.
25:
5668:Limit superior and limit inferior
1944:(which can happen if and only if
5926:{\displaystyle g_{2}\leq g_{1},}
5596:{\displaystyle \mathbb {R} ^{2}}
5496:{\displaystyle \mathbb {R} ^{2}}
3673:has no such element). However,
2662:. Together they are called the
6320:but no greatest element. Since
5883:{\displaystyle g_{1}\leq g_{2}}
4409:has at least two elements then
1154:Importantly, an upper bound of
269:, that is, it is an element of
6811:be a maximal element, for any
6462:{\displaystyle s_{1}<s_{2}}
6297:has just one maximal element,
6277:Assume for contradiction that
5626:
5614:
5526:
5514:
5125:has no upper bound in the set
4472:
4460:
4428:
4416:
4373:
4361:
4277:
4265:
4111:
4099:
4076:
4064:
4020:
4008:
3941:
3929:
3869:
3857:
3722:
3710:
3596:
3584:
3201:
3189:
3112:
3100:
3053:
3041:
2966:
2954:
2735:
2723:
2590:
2578:
2421:
2409:
2211:
2199:
1902:that is not an upper bound of
1348:
1336:
1038:
1026:
947:
935:
898:
886:
852:
840:
721:
709:
338:
326:
1:
6729:can be found (such that each
5464:{\displaystyle \mathbb {R} ,}
5436:{\displaystyle \mathbb {R} ,}
4835:, it has one maximal element.
4717:will necessarily be equal to
4677:is also a maximal element of
4511:{\displaystyle S\subseteq P.}
2250:{\displaystyle S\subseteq P.}
2174:{\displaystyle S=\{1,2,3,4\}}
1539:In the particular case where
1198:required to be an element of
986:{\displaystyle S\subseteq P.}
377:{\displaystyle S\subseteq P.}
6429:cannot be maximal, that is,
5663:Maximal and minimal elements
5658:Initial and terminal objects
5642:It has no least upper bound.
5140:{\displaystyle \mathbb {R} }
3821:itself (which of course, is
2650:, to avoid confusion with a
179:maximal and minimal elements
6607:{\displaystyle s_{2}\leq m}
6498:{\displaystyle s_{2}\in S.}
6169:is a maximal element, then
6006:is the greatest element of
5966:{\displaystyle g_{1}=g_{2}}
5564:{\displaystyle 0<x<1}
5213:{\displaystyle \{a,b,c,d\}}
4921:{\displaystyle S=\{1,2,4\}}
4637:is the greatest element of
3755:that is both comparable to
3577:comparable. Consequently,
3182:to be a maximal element of
1963:{\displaystyle u\not \in S}
641:and if it also satisfies:
462:and if it also satisfies:
7138:
7090:Cambridge University Press
6554:{\displaystyle m<s_{2}}
6422:{\displaystyle s_{1}\in S}
6366:{\displaystyle s_{1}\in S}
4592:that is also contained in
4382:{\displaystyle (R,\leq ).}
4085:{\displaystyle (P,\leq ).}
2921:By definition, an element
2484:if and only if there does
1412:is also an upper bound of
907:{\displaystyle (P,\leq ).}
7042:{\displaystyle S=\{a,b\}}
5169:{\displaystyle \,\leq \,}
5014:{\displaystyle \,\leq \,}
4853:{\displaystyle \,\leq \,}
4789:ascending chain condition
4478:{\displaystyle (P,\leq )}
4434:{\displaystyle (R,\leq )}
4283:{\displaystyle (R,\leq )}
4247:{\displaystyle i,j\in R.}
4163:{\displaystyle \,\leq \,}
4117:{\displaystyle (P,\leq )}
4026:{\displaystyle (P,\leq )}
3947:{\displaystyle (P,\leq )}
3875:{\displaystyle (P,\leq )}
3728:{\displaystyle (S,\leq )}
3602:{\displaystyle (S,\leq )}
3373:{\displaystyle \,\leq \,}
3207:{\displaystyle (P,\leq )}
3118:{\displaystyle (P,\leq )}
3059:{\displaystyle (P,\leq )}
2972:{\displaystyle (P,\leq )}
2947:is a greatest element of
2870:is true for all elements
2741:{\displaystyle (P,\leq )}
2596:{\displaystyle (P,\leq )}
2427:{\displaystyle (P,\leq )}
2217:{\displaystyle (P,\leq )}
1994:be a greatest element of
1755:is a greatest element of
1492:is a greatest element of
1354:{\displaystyle (P,\leq )}
1269:is a greatest element of
1044:{\displaystyle (P,\leq )}
953:{\displaystyle (P,\leq )}
858:{\displaystyle (P,\leq )}
727:{\displaystyle (P,\leq )}
344:{\displaystyle (P,\leq )}
221:(poset) is an element of
7011:were incomparable, then
7004:{\displaystyle a,b\in P}
6943:{\displaystyle s\leq m.}
6885:{\displaystyle m\leq s.}
6077:{\displaystyle s\leq g.}
5850:are both greatest, then
5326:{\displaystyle b\leq d.}
5298:{\displaystyle b\leq c,}
5270:{\displaystyle a\leq d,}
5242:{\displaystyle a\leq c,}
4838:When the restriction of
3703:is a maximal element of
3319:{\displaystyle s\leq m.}
3001:{\displaystyle s\leq g,}
2773:{\displaystyle x,y\in P}
2562:{\displaystyle s\neq m.}
2464:is a maximal element of
2392:{\displaystyle s\leq m.}
2363:{\displaystyle m\leq s,}
2062:have a greatest element
1650:is an element such that
1452:) but an upper bound of
6856:{\displaystyle s\leq m}
6188:{\displaystyle M\leq g}
6133:{\displaystyle g\neq s}
6107:{\displaystyle g\leq s}
6048:{\displaystyle s\in S,}
5780:{\displaystyle \geq m,}
5358:{\displaystyle \{a,b\}}
4831:has a greatest element
4209:{\displaystyle i\leq j}
3794:{\displaystyle \geq m,}
3546:{\displaystyle j\leq i}
3520:{\displaystyle i\leq j}
3474:{\displaystyle i\neq j}
3419:{\displaystyle i\leq j}
3270:{\displaystyle m\leq s}
3240:{\displaystyle s\in P,}
2863:{\displaystyle x\leq x}
2829:{\displaystyle y\leq x}
2803:{\displaystyle x\leq y}
2533:{\displaystyle m\leq s}
1724:{\displaystyle s\in S,}
1695:{\displaystyle s\leq u}
1385:{\displaystyle g\in S.}
1147:{\displaystyle s\in S.}
1118:{\displaystyle s\leq u}
830:is defined similarly.
692:{\displaystyle s\in S.}
663:{\displaystyle l\leq s}
513:{\displaystyle s\in S.}
484:{\displaystyle s\leq g}
7043:
7005:
6970:is a greatest element.
6964:
6944:
6915:
6886:
6857:
6831:
6830:{\displaystyle s\in S}
6805:
6804:{\displaystyle m\in S}
6770:
6750:
6723:
6658:
6657:{\displaystyle s_{1}.}
6628:
6608:
6575:
6555:
6522:
6499:
6463:
6423:
6390:
6367:
6340:is not greatest, some
6334:
6314:
6291:
6255:
6235:
6209:
6189:
6163:
6134:
6108:
6078:
6049:
6020:
6000:
5967:
5927:
5884:
5844:
5817:
5781:
5755:
5735:
5718:that is comparable to
5712:
5674:Upper and lower bounds
5636:
5635:{\displaystyle (1,0).}
5597:
5565:
5533:
5497:
5465:
5437:
5402:
5379:
5359:
5327:
5299:
5271:
5243:
5214:
5170:
5141:
5117:
5038:
5015:
4993:
4970:
4945:
4922:
4874:
4854:
4825:
4805:
4781:
4756:
4734:
4711:
4691:
4671:
4651:
4631:
4609:
4586:
4562:
4535:
4512:
4479:
4435:
4403:
4383:
4348:
4324:
4304:
4284:
4248:
4210:
4184:
4164:
4142:
4118:
4086:
4047:
4027:
3995:
3968:
3948:
3916:
3896:
3876:
3838:
3837:{\displaystyle \leq m}
3815:
3795:
3769:
3749:
3729:
3697:
3696:{\displaystyle m\in S}
3667:
3647:
3623:
3603:
3567:
3547:
3521:
3495:
3475:
3449:
3420:
3394:
3374:
3348:
3320:
3291:
3271:
3241:
3208:
3176:
3175:{\displaystyle m\in P}
3146:
3119:
3087:
3060:
3028:
3027:{\displaystyle s\in P}
3002:
2973:
2941:
2940:{\displaystyle g\in P}
2904:
2884:
2864:
2830:
2804:
2774:
2742:
2710:
2696:and a maximal element
2690:
2654:. The dual terms are
2627:
2597:
2563:
2534:
2508:
2507:{\displaystyle s\in S}
2478:
2458:
2457:{\displaystyle m\in S}
2428:
2393:
2364:
2335:
2334:{\displaystyle s\in S}
2303:
2277:
2276:{\displaystyle m\in S}
2251:
2218:
2182:
2175:
2101:
2080:
2052:
2032:
2018:a greatest element of
2008:
1984:
1964:
1937:
1916:
1895:
1874:
1854:
1830:
1809:
1789:
1769:
1749:
1725:
1696:
1670:
1669:{\displaystyle u\in S}
1644:
1623:
1602:
1582:
1562:
1533:
1506:
1486:
1466:
1446:
1426:
1406:
1386:
1355:
1323:
1303:
1283:
1263:
1243:
1242:{\displaystyle g\in P}
1215:
1188:
1168:
1148:
1119:
1093:
1092:{\displaystyle u\in P}
1067:
1045:
1013:
987:
954:
908:
859:
823:
797:
772:
752:
728:
693:
664:
635:
634:{\displaystyle l\in S}
607:
583:
582:{\displaystyle l\in P}
557:
537:
514:
485:
456:
455:{\displaystyle g\in S}
428:
404:
403:{\displaystyle g\in P}
378:
345:
306:
283:
255:
235:
211:
182:
171:
97:
77:
53:
7044:
7006:
6965:
6945:
6916:
6887:
6858:
6832:
6806:
6771:
6751:
6749:{\displaystyle s_{i}}
6724:
6659:
6629:
6609:
6576:
6556:
6523:
6500:
6464:
6424:
6391:
6368:
6335:
6315:
6292:
6256:
6236:
6210:
6190:
6164:
6135:
6109:
6079:
6050:
6021:
6001:
5968:
5928:
5885:
5845:
5843:{\displaystyle g_{2}}
5818:
5816:{\displaystyle g_{1}}
5782:
5756:
5741:which is necessarily
5736:
5713:
5637:
5605:lexicographical order
5598:
5566:
5534:
5532:{\displaystyle (x,y)}
5498:
5466:
5438:
5403:
5380:
5360:
5328:
5300:
5272:
5244:
5215:
5171:
5142:
5112:
5092:completeness property
5048:Sufficient conditions
5039:
5016:
4994:
4971:
4946:
4923:
4875:
4855:
4826:
4806:
4782:
4757:
4735:
4712:
4692:
4672:
4652:
4632:
4610:
4587:
4563:
4536:
4513:
4487:partially ordered set
4480:
4436:
4404:
4389:So in particular, if
4384:
4349:
4325:
4305:
4285:
4249:
4211:
4185:
4165:
4143:
4119:
4087:
4048:
4028:
3996:
3969:
3949:
3917:
3897:
3877:
3839:
3816:
3796:
3770:
3750:
3730:
3698:
3668:
3648:
3624:
3604:
3568:
3548:
3522:
3496:
3476:
3450:
3421:
3395:
3375:
3349:
3321:
3292:
3272:
3242:
3214:can be reworded as:
3209:
3177:
3147:
3120:
3088:
3061:
3029:
3003:
2974:
2942:
2905:
2885:
2865:
2831:
2805:
2775:
2743:
2711:
2691:
2628:
2626:{\displaystyle S:=P.}
2598:
2564:
2535:
2509:
2479:
2459:
2436:partially ordered set
2429:
2394:
2365:
2336:
2304:
2278:
2252:
2219:
2176:
2130:
2102:
2081:
2053:
2033:
2009:
1985:
1965:
1938:
1917:
1896:
1875:
1860:is an upper bound of
1855:
1831:
1810:
1795:is an upper bound of
1790:
1770:
1750:
1726:
1697:
1671:
1645:
1624:
1603:
1588:is an upper bound of
1583:
1563:
1534:
1507:
1487:
1467:
1447:
1427:
1407:
1387:
1356:
1324:
1309:is an upper bound of
1304:
1284:
1264:
1244:
1216:
1189:
1169:
1149:
1120:
1094:
1068:
1046:
1014:
988:
955:
909:
860:
824:
798:
773:
753:
736:partially ordered set
729:
694:
665:
636:
608:
584:
558:
538:
515:
486:
457:
429:
405:
379:
346:
307:
284:
256:
236:
219:partially ordered set
212:
172:
98:
78:
54:
35:
7015:
6983:
6954:
6925:
6914:{\displaystyle m=s,}
6896:
6867:
6841:
6815:
6789:
6760:
6733:
6668:
6638:
6618:
6585:
6581:'s maximality while
6565:
6532:
6509:
6473:
6433:
6400:
6377:
6344:
6324:
6301:
6281:
6245:
6219:
6199:
6173:
6153:
6118:
6092:
6059:
6030:
6010:
5990:
5937:
5894:
5854:
5827:
5800:
5765:
5745:
5722:
5702:
5611:
5578:
5543:
5511:
5478:
5450:
5422:
5389:
5369:
5337:
5308:
5280:
5252:
5224:
5180:
5158:
5129:
5094:of a partial order.
5088:complemented lattice
5025:
5021:is a total order on
5003:
4980:
4960:
4935:
4888:
4864:
4842:
4815:
4795:
4771:
4746:
4721:
4701:
4681:
4661:
4641:
4621:
4596:
4576:
4552:
4525:
4493:
4457:
4413:
4393:
4358:
4338:
4314:
4294:
4262:
4223:
4194:
4174:
4152:
4132:
4096:
4061:
4037:
4005:
3982:
3974:being comparable to
3958:
3926:
3906:
3886:
3854:
3825:
3805:
3779:
3759:
3739:
3707:
3681:
3657:
3637:
3613:
3581:
3557:
3531:
3505:
3485:
3459:
3448:{\displaystyle i=j.}
3430:
3404:
3384:
3362:
3354:is a set containing
3338:
3301:
3281:
3255:
3222:
3186:
3160:
3133:
3097:
3074:
3038:
3012:
2983:
2951:
2925:
2894:
2874:
2848:
2814:
2788:
2752:
2720:
2716:of a preordered set
2700:
2680:
2608:
2575:
2544:
2518:
2492:
2468:
2442:
2406:
2374:
2345:
2319:
2293:
2261:
2232:
2196:
2135:
2091:
2070:
2042:
2022:
1998:
1974:
1948:
1927:
1906:
1885:
1864:
1844:
1820:
1799:
1779:
1759:
1739:
1733:completely identical
1706:
1680:
1654:
1634:
1613:
1592:
1572:
1561:{\displaystyle P=S,}
1543:
1520:
1496:
1476:
1456:
1436:
1416:
1396:
1367:
1333:
1313:
1293:
1273:
1253:
1227:
1202:
1178:
1158:
1129:
1103:
1077:
1057:
1023:
1003:
968:
932:
883:
837:
813:
787:
783:greatest element of
762:
742:
706:
674:
648:
619:
597:
567:
547:
527:
495:
469:
440:
418:
414:greatest element of
388:
359:
323:
293:
273:
245:
225:
201:
107:
87:
67:
43:
6921:so it follows that
6756:is incomparable to
6469:must hold for some
6273:see above. —
6234:{\displaystyle M=g}
6215:is greatest, hence
5571:has no upper bound.
5507:, the set of pairs
4445:greatest elements.
4330:are comparable and
4057:maximal element of
3801:that element being
2640:totally ordered set
2571:maximal element of
1568:the definition of "
7039:
7001:
6960:
6940:
6911:
6882:
6853:
6827:
6801:
6766:
6746:
6719:
6654:
6624:
6604:
6571:
6551:
6521:{\displaystyle m,}
6518:
6495:
6459:
6419:
6389:{\displaystyle m.}
6386:
6363:
6330:
6313:{\displaystyle m,}
6310:
6287:
6251:
6231:
6205:
6185:
6159:
6130:
6104:
6074:
6045:
6016:
5996:
5963:
5923:
5880:
5840:
5813:
5777:
5751:
5734:{\displaystyle m,}
5731:
5708:
5632:
5593:
5561:
5529:
5493:
5461:
5433:
5401:{\displaystyle d,}
5398:
5375:
5355:
5323:
5295:
5267:
5239:
5210:
5166:
5137:
5118:
5037:{\displaystyle P.}
5034:
5011:
4992:{\displaystyle P,}
4989:
4966:
4941:
4918:
4870:
4850:
4821:
4801:
4777:
4752:
4733:{\displaystyle g.}
4730:
4707:
4687:
4667:
4647:
4627:
4608:{\displaystyle S.}
4605:
4582:
4558:
4531:
4508:
4475:
4431:
4399:
4379:
4344:
4320:
4300:
4280:
4244:
4206:
4190:by declaring that
4180:
4160:
4138:
4114:
4082:
4043:
4023:
3994:{\displaystyle P,}
3991:
3964:
3944:
3912:
3892:
3872:
3847:In contrast, if a
3834:
3811:
3791:
3765:
3745:
3725:
3693:
3663:
3643:
3619:
3599:
3563:
3543:
3517:
3491:
3471:
3445:
3416:
3400:by declaring that
3390:
3370:
3344:
3316:
3297:are ignored) then
3287:
3267:
3237:
3204:
3172:
3145:{\displaystyle P.}
3142:
3115:
3086:{\displaystyle P.}
3083:
3056:
3024:
2998:
2969:
2937:
2900:
2880:
2860:
2844:(which means that
2836:; they are called
2826:
2800:
2770:
2738:
2706:
2686:
2623:
2593:
2559:
2530:
2504:
2474:
2454:
2424:
2389:
2360:
2331:
2299:
2273:
2247:
2214:
2183:
2171:
2097:
2076:
2058:to simultaneously
2048:
2028:
2004:
1980:
1960:
1933:
1912:
1891:
1870:
1850:
1826:
1805:
1785:
1765:
1745:
1721:
1692:
1666:
1640:
1619:
1598:
1578:
1558:
1532:{\displaystyle S.}
1529:
1512:if and only if it
1502:
1482:
1462:
1442:
1422:
1402:
1382:
1351:
1319:
1299:
1279:
1259:
1239:
1214:{\displaystyle S.}
1211:
1184:
1164:
1144:
1115:
1089:
1063:
1041:
1009:
983:
950:
904:
855:
819:
804:. The terminology
793:
768:
748:
724:
689:
660:
631:
603:
579:
553:
533:
510:
481:
452:
424:
400:
374:
341:
305:{\displaystyle S.}
302:
279:
251:
231:
207:
183:
167:
103:". The red subset
93:
73:
49:
7099:978-0-521-78451-1
7071:topological space
6963:{\displaystyle m}
6769:{\displaystyle m}
6627:{\displaystyle m}
6574:{\displaystyle m}
6333:{\displaystyle m}
6290:{\displaystyle S}
6254:{\displaystyle M}
6208:{\displaystyle g}
6162:{\displaystyle M}
6019:{\displaystyle S}
5999:{\displaystyle g}
5754:{\displaystyle m}
5711:{\displaystyle S}
5378:{\displaystyle c}
5365:has upper bounds
5154:Let the relation
4969:{\displaystyle S}
4944:{\displaystyle S}
4873:{\displaystyle S}
4824:{\displaystyle P}
4804:{\displaystyle S}
4780:{\displaystyle P}
4755:{\displaystyle S}
4710:{\displaystyle S}
4690:{\displaystyle S}
4670:{\displaystyle g}
4650:{\displaystyle S}
4630:{\displaystyle g}
4585:{\displaystyle S}
4561:{\displaystyle S}
4541:can have at most
4534:{\displaystyle S}
4402:{\displaystyle R}
4347:{\displaystyle R}
4323:{\displaystyle R}
4303:{\displaystyle R}
4183:{\displaystyle R}
4141:{\displaystyle R}
4046:{\displaystyle g}
3967:{\displaystyle g}
3915:{\displaystyle g}
3895:{\displaystyle g}
3814:{\displaystyle m}
3768:{\displaystyle m}
3748:{\displaystyle S}
3666:{\displaystyle S}
3646:{\displaystyle S}
3622:{\displaystyle S}
3566:{\displaystyle S}
3494:{\displaystyle S}
3393:{\displaystyle S}
3347:{\displaystyle S}
3290:{\displaystyle m}
2903:{\displaystyle x}
2890:), every element
2883:{\displaystyle x}
2709:{\displaystyle m}
2689:{\displaystyle g}
2477:{\displaystyle S}
2370:then necessarily
2302:{\displaystyle S}
2117:least upper bound
2100:{\displaystyle P}
2079:{\displaystyle S}
2051:{\displaystyle S}
2031:{\displaystyle S}
2007:{\displaystyle S}
1983:{\displaystyle u}
1936:{\displaystyle S}
1915:{\displaystyle S}
1894:{\displaystyle P}
1873:{\displaystyle S}
1853:{\displaystyle u}
1829:{\displaystyle S}
1808:{\displaystyle S}
1788:{\displaystyle g}
1768:{\displaystyle S}
1748:{\displaystyle g}
1643:{\displaystyle u}
1622:{\displaystyle S}
1601:{\displaystyle S}
1581:{\displaystyle u}
1505:{\displaystyle S}
1485:{\displaystyle P}
1465:{\displaystyle S}
1445:{\displaystyle P}
1425:{\displaystyle S}
1405:{\displaystyle S}
1322:{\displaystyle S}
1302:{\displaystyle g}
1282:{\displaystyle S}
1262:{\displaystyle g}
1187:{\displaystyle P}
1167:{\displaystyle S}
1066:{\displaystyle u}
1012:{\displaystyle S}
822:{\displaystyle S}
809:least element of
796:{\displaystyle S}
771:{\displaystyle S}
751:{\displaystyle S}
606:{\displaystyle S}
593:least element of
556:{\displaystyle S}
536:{\displaystyle s}
427:{\displaystyle S}
282:{\displaystyle S}
254:{\displaystyle S}
234:{\displaystyle S}
210:{\displaystyle S}
96:{\displaystyle y}
76:{\displaystyle x}
52:{\displaystyle P}
16:(Redirected from
7129:
7103:
7088:(2nd ed.).
7074:
7067:
7050:
7048:
7046:
7045:
7040:
7010:
7008:
7007:
7002:
6977:
6971:
6969:
6967:
6966:
6961:
6950:In other words,
6949:
6947:
6946:
6941:
6920:
6918:
6917:
6912:
6891:
6889:
6888:
6883:
6862:
6860:
6859:
6854:
6836:
6834:
6833:
6828:
6810:
6808:
6807:
6802:
6783:
6777:
6775:
6773:
6772:
6767:
6755:
6753:
6752:
6747:
6745:
6744:
6728:
6726:
6725:
6720:
6712:
6711:
6693:
6692:
6680:
6679:
6663:
6661:
6660:
6655:
6650:
6649:
6633:
6631:
6630:
6625:
6613:
6611:
6610:
6605:
6597:
6596:
6580:
6578:
6577:
6572:
6560:
6558:
6557:
6552:
6550:
6549:
6527:
6525:
6524:
6519:
6504:
6502:
6501:
6496:
6485:
6484:
6468:
6466:
6465:
6460:
6458:
6457:
6445:
6444:
6428:
6426:
6425:
6420:
6412:
6411:
6395:
6393:
6392:
6387:
6372:
6370:
6369:
6364:
6356:
6355:
6339:
6337:
6336:
6331:
6319:
6317:
6316:
6311:
6296:
6294:
6293:
6288:
6268:
6262:
6260:
6258:
6257:
6252:
6240:
6238:
6237:
6232:
6214:
6212:
6211:
6206:
6194:
6192:
6191:
6186:
6168:
6166:
6165:
6160:
6147:
6141:
6139:
6137:
6136:
6131:
6113:
6111:
6110:
6105:
6088:, this renders (
6083:
6081:
6080:
6075:
6054:
6052:
6051:
6046:
6025:
6023:
6022:
6017:
6005:
6003:
6002:
5997:
5984:
5978:
5972:
5970:
5969:
5964:
5962:
5961:
5949:
5948:
5932:
5930:
5929:
5924:
5919:
5918:
5906:
5905:
5889:
5887:
5886:
5881:
5879:
5878:
5866:
5865:
5849:
5847:
5846:
5841:
5839:
5838:
5822:
5820:
5819:
5814:
5812:
5811:
5794:
5788:
5787:" was redundant.
5786:
5784:
5783:
5778:
5760:
5758:
5757:
5752:
5740:
5738:
5737:
5732:
5717:
5715:
5714:
5709:
5696:
5641:
5639:
5638:
5633:
5602:
5600:
5599:
5594:
5592:
5591:
5586:
5570:
5568:
5567:
5562:
5538:
5536:
5535:
5530:
5502:
5500:
5499:
5494:
5492:
5491:
5486:
5470:
5468:
5467:
5462:
5457:
5442:
5440:
5439:
5434:
5429:
5413:rational numbers
5407:
5405:
5404:
5399:
5384:
5382:
5381:
5376:
5364:
5362:
5361:
5356:
5332:
5330:
5329:
5324:
5304:
5302:
5301:
5296:
5276:
5274:
5273:
5268:
5248:
5246:
5245:
5240:
5219:
5217:
5216:
5211:
5175:
5173:
5172:
5167:
5146:
5144:
5143:
5138:
5136:
5043:
5041:
5040:
5035:
5020:
5018:
5017:
5012:
4998:
4996:
4995:
4990:
4975:
4973:
4972:
4967:
4950:
4948:
4947:
4942:
4927:
4925:
4924:
4919:
4879:
4877:
4876:
4871:
4859:
4857:
4856:
4851:
4830:
4828:
4827:
4822:
4810:
4808:
4807:
4802:
4786:
4784:
4783:
4778:
4761:
4759:
4758:
4753:
4739:
4737:
4736:
4731:
4716:
4714:
4713:
4708:
4696:
4694:
4693:
4688:
4676:
4674:
4673:
4668:
4656:
4654:
4653:
4648:
4636:
4634:
4633:
4628:
4614:
4612:
4611:
4606:
4591:
4589:
4588:
4583:
4567:
4565:
4564:
4559:
4540:
4538:
4537:
4532:
4517:
4515:
4514:
4509:
4484:
4482:
4481:
4476:
4453:Throughout, let
4440:
4438:
4437:
4432:
4408:
4406:
4405:
4400:
4388:
4386:
4385:
4380:
4353:
4351:
4350:
4345:
4329:
4327:
4326:
4321:
4309:
4307:
4306:
4301:
4289:
4287:
4286:
4281:
4253:
4251:
4250:
4245:
4215:
4213:
4212:
4207:
4189:
4187:
4186:
4181:
4169:
4167:
4166:
4161:
4147:
4145:
4144:
4139:
4123:
4121:
4120:
4115:
4091:
4089:
4088:
4083:
4052:
4050:
4049:
4044:
4032:
4030:
4029:
4024:
4000:
3998:
3997:
3992:
3973:
3971:
3970:
3965:
3953:
3951:
3950:
3945:
3921:
3919:
3918:
3913:
3901:
3899:
3898:
3893:
3881:
3879:
3878:
3873:
3843:
3841:
3840:
3835:
3820:
3818:
3817:
3812:
3800:
3798:
3797:
3792:
3774:
3772:
3771:
3766:
3754:
3752:
3751:
3746:
3734:
3732:
3731:
3726:
3702:
3700:
3699:
3694:
3672:
3670:
3669:
3664:
3652:
3650:
3649:
3644:
3628:
3626:
3625:
3620:
3608:
3606:
3605:
3600:
3572:
3570:
3569:
3564:
3552:
3550:
3549:
3544:
3526:
3524:
3523:
3518:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3454:
3452:
3451:
3446:
3425:
3423:
3422:
3417:
3399:
3397:
3396:
3391:
3379:
3377:
3376:
3371:
3353:
3351:
3350:
3345:
3325:
3323:
3322:
3317:
3296:
3294:
3293:
3288:
3276:
3274:
3273:
3268:
3246:
3244:
3243:
3238:
3213:
3211:
3210:
3205:
3181:
3179:
3178:
3173:
3151:
3149:
3148:
3143:
3124:
3122:
3121:
3116:
3092:
3090:
3089:
3084:
3065:
3063:
3062:
3057:
3033:
3031:
3030:
3025:
3007:
3005:
3004:
2999:
2978:
2976:
2975:
2970:
2946:
2944:
2943:
2938:
2909:
2907:
2906:
2901:
2889:
2887:
2886:
2881:
2869:
2867:
2866:
2861:
2835:
2833:
2832:
2827:
2809:
2807:
2806:
2801:
2779:
2777:
2776:
2771:
2747:
2745:
2744:
2739:
2715:
2713:
2712:
2707:
2695:
2693:
2692:
2687:
2665:absolute extrema
2660:absolute minimum
2648:absolute maximum
2632:
2630:
2629:
2624:
2602:
2600:
2599:
2594:
2568:
2566:
2565:
2560:
2539:
2537:
2536:
2531:
2513:
2511:
2510:
2505:
2483:
2481:
2480:
2475:
2463:
2461:
2460:
2455:
2433:
2431:
2430:
2425:
2398:
2396:
2395:
2390:
2369:
2367:
2366:
2361:
2340:
2338:
2337:
2332:
2308:
2306:
2305:
2300:
2283:is said to be a
2282:
2280:
2279:
2274:
2256:
2254:
2253:
2248:
2223:
2221:
2220:
2215:
2180:
2178:
2177:
2172:
2106:
2104:
2103:
2098:
2085:
2083:
2082:
2077:
2057:
2055:
2054:
2049:
2037:
2035:
2034:
2029:
2013:
2011:
2010:
2005:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1942:
1940:
1939:
1934:
1921:
1919:
1918:
1913:
1900:
1898:
1897:
1892:
1879:
1877:
1876:
1871:
1859:
1857:
1856:
1851:
1835:
1833:
1832:
1827:
1814:
1812:
1811:
1806:
1794:
1792:
1791:
1786:
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1730:
1728:
1727:
1722:
1701:
1699:
1698:
1693:
1675:
1673:
1672:
1667:
1649:
1647:
1646:
1641:
1628:
1626:
1625:
1620:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1567:
1565:
1564:
1559:
1538:
1536:
1535:
1530:
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1471:
1469:
1468:
1463:
1451:
1449:
1448:
1443:
1431:
1429:
1428:
1423:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1360:
1358:
1357:
1352:
1328:
1326:
1325:
1320:
1308:
1306:
1305:
1300:
1288:
1286:
1285:
1280:
1268:
1266:
1265:
1260:
1248:
1246:
1245:
1240:
1220:
1218:
1217:
1212:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1153:
1151:
1150:
1145:
1124:
1122:
1121:
1116:
1098:
1096:
1095:
1090:
1072:
1070:
1069:
1064:
1050:
1048:
1047:
1042:
1018:
1016:
1015:
1010:
992:
990:
989:
984:
959:
957:
956:
951:
913:
911:
910:
905:
864:
862:
861:
856:
828:
826:
825:
820:
802:
800:
799:
794:
777:
775:
774:
769:
757:
755:
754:
749:
733:
731:
730:
725:
698:
696:
695:
690:
669:
667:
666:
661:
640:
638:
637:
632:
612:
610:
609:
604:
588:
586:
585:
580:
562:
560:
559:
554:
542:
540:
539:
534:
519:
517:
516:
511:
490:
488:
487:
482:
461:
459:
458:
453:
433:
431:
430:
425:
409:
407:
406:
401:
383:
381:
380:
375:
350:
348:
347:
342:
311:
309:
308:
303:
288:
286:
285:
280:
260:
258:
257:
252:
240:
238:
237:
232:
216:
214:
213:
208:
195:greatest element
189:, especially in
176:
174:
173:
168:
102:
100:
99:
94:
82:
80:
79:
74:
58:
56:
55:
50:
21:
7137:
7136:
7132:
7131:
7130:
7128:
7127:
7126:
7107:
7106:
7100:
7081:
7078:
7077:
7068:
7064:
7059:
7054:
7053:
7013:
7012:
6981:
6980:
6978:
6974:
6952:
6951:
6923:
6922:
6894:
6893:
6865:
6864:
6839:
6838:
6813:
6812:
6787:
6786:
6784:
6780:
6758:
6757:
6736:
6731:
6730:
6703:
6684:
6671:
6666:
6665:
6641:
6636:
6635:
6616:
6615:
6588:
6583:
6582:
6563:
6562:
6541:
6530:
6529:
6507:
6506:
6476:
6471:
6470:
6449:
6436:
6431:
6430:
6403:
6398:
6397:
6375:
6374:
6347:
6342:
6341:
6322:
6321:
6299:
6298:
6279:
6278:
6269:
6265:
6243:
6242:
6217:
6216:
6197:
6196:
6171:
6170:
6151:
6150:
6148:
6144:
6116:
6115:
6090:
6089:
6057:
6056:
6028:
6027:
6008:
6007:
5988:
5987:
5985:
5981:
5953:
5940:
5935:
5934:
5910:
5897:
5892:
5891:
5870:
5857:
5852:
5851:
5830:
5825:
5824:
5803:
5798:
5797:
5795:
5791:
5763:
5762:
5743:
5742:
5720:
5719:
5700:
5699:
5697:
5693:
5688:
5670:(infimum limit)
5649:
5609:
5608:
5581:
5576:
5575:
5541:
5540:
5509:
5508:
5481:
5476:
5475:
5448:
5447:
5420:
5419:
5387:
5386:
5367:
5366:
5335:
5334:
5306:
5305:
5278:
5277:
5250:
5249:
5222:
5221:
5178:
5177:
5156:
5155:
5127:
5126:
5107:
5064:
5050:
5023:
5022:
5001:
5000:
4978:
4977:
4958:
4957:
4933:
4932:
4886:
4885:
4862:
4861:
4840:
4839:
4833:if, and only if
4813:
4812:
4793:
4792:
4769:
4768:
4744:
4743:
4719:
4718:
4699:
4698:
4679:
4678:
4659:
4658:
4639:
4638:
4619:
4618:
4594:
4593:
4574:
4573:
4550:
4549:
4523:
4522:
4491:
4490:
4455:
4454:
4451:
4411:
4410:
4391:
4390:
4356:
4355:
4336:
4335:
4312:
4311:
4292:
4291:
4260:
4259:
4258:preordered set
4221:
4220:
4192:
4191:
4172:
4171:
4150:
4149:
4130:
4129:
4094:
4093:
4059:
4058:
4035:
4034:
4003:
4002:
3980:
3979:
3956:
3955:
3924:
3923:
3904:
3903:
3884:
3883:
3852:
3851:
3823:
3822:
3803:
3802:
3777:
3776:
3757:
3756:
3737:
3736:
3705:
3704:
3679:
3678:
3655:
3654:
3635:
3634:
3611:
3610:
3579:
3578:
3555:
3554:
3529:
3528:
3503:
3502:
3483:
3482:
3457:
3456:
3428:
3427:
3426:if and only if
3402:
3401:
3382:
3381:
3360:
3359:
3336:
3335:
3299:
3298:
3279:
3278:
3253:
3252:
3220:
3219:
3184:
3183:
3158:
3157:
3131:
3130:
3095:
3094:
3072:
3071:
3036:
3035:
3010:
3009:
2981:
2980:
2949:
2948:
2923:
2922:
2892:
2891:
2872:
2871:
2846:
2845:
2812:
2811:
2786:
2785:
2780:are said to be
2750:
2749:
2718:
2717:
2698:
2697:
2678:
2677:
2606:
2605:
2573:
2572:
2542:
2541:
2516:
2515:
2490:
2489:
2466:
2465:
2440:
2439:
2404:
2403:
2372:
2371:
2343:
2342:
2317:
2316:
2291:
2290:
2287:maximal element
2259:
2258:
2230:
2229:
2194:
2193:
2187:maximal element
2133:
2132:
2125:
2089:
2088:
2068:
2067:
2040:
2039:
2020:
2019:
1996:
1995:
1972:
1971:
1946:
1945:
1925:
1924:
1904:
1903:
1883:
1882:
1862:
1861:
1842:
1841:
1818:
1817:
1797:
1796:
1777:
1776:
1775:if and only if
1757:
1756:
1737:
1736:
1704:
1703:
1678:
1677:
1652:
1651:
1632:
1631:
1611:
1610:
1590:
1589:
1570:
1569:
1541:
1540:
1518:
1517:
1494:
1493:
1474:
1473:
1454:
1453:
1434:
1433:
1414:
1413:
1394:
1393:
1365:
1364:
1331:
1330:
1311:
1310:
1291:
1290:
1289:if and only if
1271:
1270:
1251:
1250:
1225:
1224:
1200:
1199:
1176:
1175:
1156:
1155:
1127:
1126:
1101:
1100:
1075:
1074:
1055:
1054:
1021:
1020:
1001:
1000:
966:
965:
930:
929:
919:
881:
880:
835:
834:
811:
810:
785:
784:
760:
759:
740:
739:
704:
703:
672:
671:
646:
645:
617:
616:
595:
594:
565:
564:
545:
544:
525:
524:
493:
492:
467:
466:
438:
437:
416:
415:
386:
385:
357:
356:
321:
320:
317:
291:
290:
271:
270:
243:
242:
223:
222:
199:
198:
105:
104:
85:
84:
65:
64:
41:
40:
30:
23:
22:
15:
12:
11:
5:
7135:
7133:
7125:
7124:
7119:
7109:
7108:
7105:
7104:
7098:
7076:
7075:
7061:
7060:
7058:
7055:
7052:
7051:
7038:
7035:
7032:
7029:
7026:
7023:
7020:
7000:
6997:
6994:
6991:
6988:
6972:
6959:
6939:
6936:
6933:
6930:
6910:
6907:
6904:
6901:
6881:
6878:
6875:
6872:
6852:
6849:
6846:
6826:
6823:
6820:
6800:
6797:
6794:
6778:
6765:
6743:
6739:
6718:
6715:
6710:
6706:
6702:
6699:
6696:
6691:
6687:
6683:
6678:
6674:
6653:
6648:
6644:
6623:
6603:
6600:
6595:
6591:
6570:
6548:
6544:
6540:
6537:
6517:
6514:
6494:
6491:
6488:
6483:
6479:
6456:
6452:
6448:
6443:
6439:
6418:
6415:
6410:
6406:
6385:
6382:
6362:
6359:
6354:
6350:
6329:
6309:
6306:
6286:
6263:
6250:
6230:
6227:
6224:
6204:
6184:
6181:
6178:
6158:
6142:
6129:
6126:
6123:
6103:
6100:
6097:
6073:
6070:
6067:
6064:
6044:
6041:
6038:
6035:
6015:
5995:
5979:
5960:
5956:
5952:
5947:
5943:
5922:
5917:
5913:
5909:
5904:
5900:
5877:
5873:
5869:
5864:
5860:
5837:
5833:
5810:
5806:
5789:
5776:
5773:
5770:
5750:
5730:
5727:
5707:
5690:
5689:
5687:
5684:
5683:
5682:
5676:
5671:
5665:
5660:
5655:
5648:
5645:
5644:
5643:
5631:
5628:
5625:
5622:
5619:
5616:
5590:
5585:
5572:
5560:
5557:
5554:
5551:
5548:
5528:
5525:
5522:
5519:
5516:
5490:
5485:
5472:
5460:
5456:
5444:
5432:
5428:
5416:
5409:
5397:
5394:
5374:
5354:
5351:
5348:
5345:
5342:
5322:
5319:
5316:
5313:
5294:
5291:
5288:
5285:
5266:
5263:
5260:
5257:
5238:
5235:
5232:
5229:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5164:
5152:
5135:
5121:The subset of
5106:
5103:
5063:
5062:Top and bottom
5060:
5059:
5058:
5049:
5046:
5045:
5044:
5033:
5030:
5009:
4988:
4985:
4965:
4954:
4953:
4952:
4940:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4869:
4848:
4836:
4820:
4800:
4787:satisfies the
4776:
4765:
4764:
4763:
4751:
4742:Thus if a set
4729:
4726:
4706:
4686:
4666:
4646:
4626:
4615:
4604:
4601:
4581:
4557:
4546:
4544:
4530:
4507:
4504:
4501:
4498:
4474:
4471:
4468:
4465:
4462:
4450:
4447:
4444:
4430:
4427:
4424:
4421:
4418:
4398:
4378:
4375:
4372:
4369:
4366:
4363:
4343:
4333:
4319:
4299:
4279:
4276:
4273:
4270:
4267:
4243:
4240:
4237:
4234:
4231:
4228:
4219:holds for all
4218:
4205:
4202:
4199:
4179:
4158:
4137:
4127:
4113:
4110:
4107:
4104:
4101:
4081:
4078:
4075:
4072:
4069:
4066:
4056:
4042:
4022:
4019:
4016:
4013:
4010:
3990:
3987:
3977:
3963:
3943:
3940:
3937:
3934:
3931:
3911:
3891:
3871:
3868:
3865:
3862:
3859:
3849:preordered set
3833:
3830:
3810:
3790:
3787:
3784:
3764:
3744:
3724:
3721:
3718:
3715:
3712:
3692:
3689:
3686:
3676:
3662:
3642:
3632:
3618:
3598:
3595:
3592:
3589:
3586:
3576:
3562:
3542:
3539:
3536:
3516:
3513:
3510:
3490:
3470:
3467:
3464:
3444:
3441:
3438:
3435:
3415:
3412:
3409:
3389:
3368:
3357:
3343:
3332:
3331:
3327:
3326:
3315:
3312:
3309:
3306:
3286:
3266:
3263:
3260:
3250:
3236:
3233:
3230:
3227:
3203:
3200:
3197:
3194:
3191:
3171:
3168:
3165:
3155:
3141:
3138:
3128:
3114:
3111:
3108:
3105:
3102:
3082:
3079:
3069:
3055:
3052:
3049:
3046:
3043:
3023:
3020:
3017:
2997:
2994:
2991:
2988:
2968:
2965:
2962:
2959:
2956:
2936:
2933:
2930:
2913:
2899:
2879:
2859:
2856:
2853:
2839:
2825:
2822:
2819:
2799:
2796:
2793:
2783:
2769:
2766:
2763:
2760:
2757:
2737:
2734:
2731:
2728:
2725:
2705:
2685:
2674:
2673:
2622:
2619:
2616:
2613:
2603:
2592:
2589:
2586:
2583:
2580:
2558:
2555:
2552:
2549:
2529:
2526:
2523:
2503:
2500:
2497:
2487:
2473:
2453:
2450:
2447:
2423:
2420:
2417:
2414:
2411:
2400:
2399:
2388:
2385:
2382:
2379:
2359:
2356:
2353:
2350:
2330:
2327:
2324:
2309:
2298:
2272:
2269:
2266:
2246:
2243:
2240:
2237:
2226:preordered set
2213:
2210:
2207:
2204:
2201:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2124:
2121:
2107:
2096:
2075:
2065:
2061:
2047:
2027:
2017:
2003:
1993:
1979:
1959:
1956:
1953:
1943:
1932:
1911:
1901:
1890:
1869:
1849:
1836:
1825:
1804:
1784:
1764:
1744:
1734:
1720:
1717:
1714:
1711:
1691:
1688:
1685:
1665:
1662:
1659:
1639:
1629:
1618:
1597:
1577:
1557:
1554:
1551:
1548:
1528:
1525:
1515:
1501:
1481:
1461:
1441:
1421:
1401:
1381:
1378:
1375:
1372:
1363:
1350:
1347:
1344:
1341:
1338:
1318:
1298:
1278:
1258:
1238:
1235:
1232:
1210:
1207:
1197:
1183:
1163:
1143:
1140:
1137:
1134:
1114:
1111:
1108:
1088:
1085:
1082:
1062:
1053:is an element
1051:
1040:
1037:
1034:
1031:
1028:
1008:
982:
979:
976:
973:
962:preordered set
949:
946:
943:
940:
937:
918:
915:
903:
900:
897:
894:
891:
888:
878:
871:
854:
851:
848:
845:
842:
818:
808:
792:
782:
767:
747:
723:
720:
717:
714:
711:
700:
699:
688:
685:
682:
679:
659:
656:
653:
630:
627:
624:
614:
602:
589:is said to be
578:
575:
572:
552:
532:
521:
520:
509:
506:
503:
500:
480:
477:
474:
451:
448:
445:
435:
423:
410:is said to be
399:
396:
393:
373:
370:
367:
364:
353:preordered set
340:
337:
334:
331:
328:
316:
313:
301:
298:
278:
250:
230:
206:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
121:
118:
115:
112:
92:
72:
48:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7134:
7123:
7120:
7118:
7115:
7114:
7112:
7101:
7095:
7091:
7087:
7086:
7080:
7079:
7072:
7066:
7063:
7056:
7033:
7030:
7027:
7021:
7018:
6998:
6995:
6992:
6989:
6986:
6976:
6973:
6957:
6937:
6934:
6931:
6928:
6908:
6905:
6902:
6899:
6879:
6876:
6873:
6870:
6850:
6847:
6844:
6824:
6821:
6818:
6798:
6795:
6792:
6782:
6779:
6763:
6741:
6737:
6716:
6713:
6708:
6704:
6700:
6697:
6694:
6689:
6685:
6681:
6676:
6672:
6651:
6646:
6642:
6621:
6601:
6598:
6593:
6589:
6568:
6546:
6542:
6538:
6535:
6515:
6512:
6492:
6489:
6486:
6481:
6477:
6454:
6450:
6446:
6441:
6437:
6416:
6413:
6408:
6404:
6383:
6380:
6360:
6357:
6352:
6348:
6327:
6307:
6304:
6284:
6276:
6272:
6267:
6264:
6248:
6228:
6225:
6222:
6202:
6182:
6179:
6176:
6156:
6146:
6143:
6140:) impossible.
6127:
6124:
6121:
6101:
6098:
6095:
6087:
6071:
6068:
6065:
6062:
6042:
6039:
6036:
6033:
6013:
5993:
5983:
5980:
5976:
5958:
5954:
5950:
5945:
5941:
5920:
5915:
5911:
5907:
5902:
5898:
5875:
5871:
5867:
5862:
5858:
5835:
5831:
5808:
5804:
5793:
5790:
5774:
5771:
5768:
5748:
5728:
5725:
5705:
5695:
5692:
5685:
5680:
5677:
5675:
5672:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5650:
5646:
5629:
5623:
5620:
5617:
5606:
5588:
5573:
5558:
5555:
5552:
5549:
5546:
5523:
5520:
5517:
5506:
5505:product order
5488:
5473:
5458:
5445:
5430:
5417:
5414:
5410:
5395:
5392:
5372:
5349:
5346:
5343:
5320:
5317:
5314:
5311:
5292:
5289:
5286:
5283:
5264:
5261:
5258:
5255:
5236:
5233:
5230:
5227:
5204:
5201:
5198:
5195:
5192:
5189:
5186:
5162:
5153:
5150:
5124:
5120:
5119:
5115:
5114:Hasse diagram
5111:
5104:
5102:
5100:
5095:
5093:
5089:
5085:
5084:bounded poset
5081:
5077:
5073:
5069:
5061:
5056:
5052:
5051:
5047:
5031:
5028:
5007:
4986:
4983:
4963:
4955:
4938:
4930:
4929:
4912:
4909:
4906:
4903:
4900:
4894:
4891:
4883:
4867:
4846:
4837:
4834:
4818:
4798:
4790:
4774:
4766:
4749:
4741:
4740:
4727:
4724:
4704:
4684:
4664:
4644:
4624:
4616:
4602:
4599:
4579:
4571:
4555:
4547:
4542:
4528:
4520:
4519:
4518:
4505:
4502:
4499:
4496:
4488:
4469:
4466:
4463:
4448:
4446:
4442:
4441:has multiple
4425:
4422:
4419:
4396:
4376:
4370:
4367:
4364:
4341:
4331:
4317:
4297:
4274:
4271:
4268:
4257:
4241:
4238:
4235:
4232:
4229:
4226:
4216:
4203:
4200:
4197:
4177:
4156:
4135:
4125:
4108:
4105:
4102:
4079:
4073:
4070:
4067:
4054:
4040:
4017:
4014:
4011:
3988:
3985:
3975:
3961:
3938:
3935:
3932:
3909:
3889:
3866:
3863:
3860:
3850:
3845:
3831:
3828:
3808:
3788:
3785:
3782:
3762:
3742:
3719:
3716:
3713:
3690:
3687:
3684:
3674:
3660:
3640:
3630:
3616:
3593:
3590:
3587:
3574:
3560:
3540:
3537:
3534:
3514:
3511:
3508:
3501:then neither
3488:
3468:
3465:
3462:
3442:
3439:
3436:
3433:
3413:
3410:
3407:
3387:
3366:
3355:
3341:
3334:Suppose that
3329:
3328:
3313:
3310:
3307:
3304:
3284:
3264:
3261:
3258:
3251:
3248:
3234:
3231:
3228:
3225:
3217:
3216:
3215:
3198:
3195:
3192:
3169:
3166:
3163:
3153:
3139:
3136:
3126:
3109:
3106:
3103:
3080:
3077:
3067:
3050:
3047:
3044:
3021:
3018:
3015:
2995:
2992:
2989:
2986:
2963:
2960:
2957:
2934:
2931:
2928:
2919:
2917:
2911:
2897:
2877:
2857:
2854:
2851:
2843:
2837:
2823:
2820:
2817:
2797:
2794:
2791:
2781:
2767:
2764:
2761:
2758:
2755:
2732:
2729:
2726:
2703:
2683:
2671:
2670:
2669:
2667:
2666:
2661:
2657:
2653:
2652:local maximum
2649:
2645:
2641:
2636:
2633:
2620:
2617:
2614:
2611:
2587:
2584:
2581:
2570:
2556:
2553:
2550:
2547:
2527:
2524:
2521:
2501:
2498:
2495:
2485:
2471:
2451:
2448:
2445:
2437:
2418:
2415:
2412:
2386:
2383:
2380:
2377:
2357:
2354:
2351:
2348:
2328:
2325:
2322:
2314:
2313:
2312:
2310:
2296:
2288:
2285:
2270:
2267:
2264:
2244:
2241:
2238:
2235:
2227:
2208:
2205:
2202:
2190:
2188:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2141:
2138:
2129:
2122:
2120:
2118:
2114:
2109:
2094:
2086:
2073:
2063:
2059:
2045:
2025:
2015:
2001:
1991:
1977:
1957:
1954:
1951:
1930:
1922:
1909:
1888:
1880:
1867:
1847:
1838:
1823:
1815:
1802:
1782:
1762:
1742:
1732:
1718:
1715:
1712:
1709:
1689:
1686:
1683:
1663:
1660:
1657:
1637:
1616:
1608:
1595:
1575:
1555:
1552:
1549:
1546:
1526:
1523:
1513:
1499:
1479:
1459:
1439:
1419:
1399:
1379:
1376:
1373:
1370:
1361:
1345:
1342:
1339:
1316:
1296:
1276:
1256:
1236:
1233:
1230:
1221:
1208:
1205:
1195:
1181:
1161:
1141:
1138:
1135:
1132:
1112:
1109:
1106:
1086:
1083:
1080:
1060:
1052:
1035:
1032:
1029:
1006:
998:
995:
980:
977:
974:
971:
963:
944:
941:
938:
926:
924:
916:
914:
901:
895:
892:
889:
877:
873:
870:
866:
849:
846:
843:
831:
829:
816:
806:
803:
790:
780:
765:
745:
737:
718:
715:
712:
686:
683:
680:
677:
657:
654:
651:
644:
643:
642:
628:
625:
622:
613:
600:
590:
576:
573:
570:
550:
530:
507:
504:
501:
498:
478:
475:
472:
465:
464:
463:
449:
446:
443:
434:
421:
411:
397:
394:
391:
371:
368:
365:
362:
354:
335:
332:
329:
314:
312:
299:
296:
276:
268:
264:
263:least element
248:
228:
220:
204:
196:
192:
188:
180:
161:
158:
155:
152:
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
113:
110:
90:
70:
62:
46:
38:
37:Hasse diagram
34:
27:
19:
18:Least element
7122:Superlatives
7117:Order theory
7084:
7065:
6975:
6781:
6561:contradicts
6274:
6270:
6266:
6145:
6086:antisymmetry
5982:
5975:antisymmetry
5792:
5694:
5220:be given by
5149:real numbers
5116:of example 2
5099:order theory
5096:
5083:
5079:
5075:
5071:
5067:
5065:
4452:
3846:
3356:at least two
3333:
3247:
2920:
2838:incomparable
2675:
2663:
2659:
2655:
2647:
2643:
2637:
2634:
2401:
2284:
2191:
2184:
2113:real numbers
2110:
1839:
1222:
994:
927:
923:upper bounds
920:
875:
868:
832:
805:
779:
701:
592:
522:
413:
318:
262:
197:of a subset
194:
191:order theory
184:
26:
6528:too, since
6261:is maximal.
4882:total order
4791:, a subset
4570:upper bound
4334:element of
3978:element of
3633:element of
3070:element in
2257:An element
1630:" becomes:
997:upper bound
384:An element
315:Definitions
265:is defined
261:. The term
187:mathematics
39:of the set
7111:Categories
7057:References
5933:and hence
5679:Well-order
4449:Properties
3481:belong to
3008:for every
2782:comparable
2514:such that
2488:exist any
2341:satisfies
1073:such that
734:is also a
6996:∈
6932:≤
6874:≤
6848:≤
6822:∈
6796:∈
6717:⋯
6698:⋯
6599:≤
6487:∈
6414:∈
6358:∈
6180:≤
6125:≠
6099:≤
6066:≤
6037:∈
5908:≤
5868:≤
5769:≥
5603:with the
5503:with the
5315:≤
5287:≤
5259:≤
5231:≤
5163:≤
5053:A finite
5008:≤
4847:≤
4500:⊆
4470:≤
4426:≤
4371:≤
4275:≤
4236:∈
4201:≤
4157:≤
4109:≤
4074:≤
4018:≤
3939:≤
3867:≤
3829:≤
3783:≥
3720:≤
3688:∈
3594:≤
3538:≤
3512:≤
3466:≠
3411:≤
3367:≤
3308:≤
3262:≤
3229:∈
3199:≤
3167:∈
3110:≤
3051:≤
3019:∈
2990:≤
2964:≤
2932:∈
2855:≤
2842:reflexive
2821:≤
2795:≤
2765:∈
2733:≤
2588:≤
2551:≠
2525:≤
2499:∈
2449:∈
2419:≤
2381:≤
2352:≤
2326:∈
2315:whenever
2268:∈
2239:⊆
2209:≤
1731:which is
1713:∈
1687:≤
1661:∈
1374:∈
1346:≤
1234:∈
1136:∈
1110:≤
1084:∈
1036:≤
975:⊆
945:≤
896:≤
850:≤
719:≤
681:∈
655:≤
626:∈
574:∈
502:∈
476:≤
447:∈
395:∈
366:⊆
336:≤
6271:Only if:
5647:See also
5333:The set
5123:integers
5105:Examples
5078:(0) and
5074:(⊤), or
5070:(⊥) and
4489:and let
4443:distinct
4256:directed
3677:element
3218:For all
2916:directed
2912:distinct
2228:and let
1955:∉
1702:for all
1125:for all
964:and let
670:for all
491:for all
355:and let
83:divides
61:divisors
6837:either
5411:In the
4053:is the
2656:minimum
2644:maximum
1970:) then
1514:belongs
872:(resp.
7096:
6396:Hence
6241:since
6195:since
5068:bottom
4568:is an
4521:A set
4217:always
876:bottom
267:dually
193:, the
6055:then
5686:Notes
5539:with
5055:chain
4999:then
4880:is a
4657:then
4485:be a
4332:every
3976:every
3902:then
3675:every
3631:every
3068:every
2638:In a
2438:then
2434:is a
2224:be a
1249:then
1099:and
960:be a
879:) of
738:then
351:be a
217:of a
7094:ISBN
6785:Let
6714:<
6701:<
6695:<
6682:<
6634:and
6539:<
6447:<
6114:and
6026:and
5890:and
5823:and
5556:<
5550:<
5385:and
5080:unit
5076:zero
4254:The
4055:only
3775:and
3653:but
3573:are
3527:nor
3125:are
2658:and
2540:and
2192:Let
1990:can
1676:and
1432:(in
928:Let
319:Let
6979:If
6863:or
6275:If:
6149:If
6084:By
5986:If
5973:by
5796:If
5574:In
5474:In
5446:In
5418:In
5176:on
5147:of
5072:top
4976:of
4860:to
4811:of
4767:If
4617:If
4572:of
4543:one
4170:on
4126:not
4124:is
4001:if
3844:).
3455:If
3380:on
3127:not
2979:if
2810:or
2784:if
2486:not
2402:If
2289:of
2108:.
2087:in
2064:and
2060:not
1992:not
1923:in
1881:in
1840:If
1837:.
1816:in
1609:in
1516:to
1472:in
1362:and
1329:in
1223:If
1196:not
1194:is
1174:in
1019:in
999:of
993:An
925:.
869:top
833:If
807:the
781:the
702:If
615:if
436:if
185:In
59:of
7113::
7092:.
5101:.
3575:in
3249:IF
3154:if
2615::=
2569:A
2016:is
874:a
867:a
591:a
412:a
162:30
156:15
150:10
7102:.
7073:.
7037:}
7034:b
7031:,
7028:a
7025:{
7022:=
7019:S
6999:P
6993:b
6990:,
6987:a
6958:m
6938:.
6935:m
6929:s
6909:,
6906:s
6903:=
6900:m
6880:.
6877:s
6871:m
6851:m
6845:s
6825:S
6819:s
6799:S
6793:m
6764:m
6742:i
6738:s
6709:n
6705:s
6690:2
6686:s
6677:1
6673:s
6652:.
6647:1
6643:s
6622:m
6602:m
6594:2
6590:s
6569:m
6547:2
6543:s
6536:m
6516:,
6513:m
6493:.
6490:S
6482:2
6478:s
6455:2
6451:s
6442:1
6438:s
6417:S
6409:1
6405:s
6384:.
6381:m
6361:S
6353:1
6349:s
6328:m
6308:,
6305:m
6285:S
6249:M
6229:g
6226:=
6223:M
6203:g
6183:g
6177:M
6157:M
6128:s
6122:g
6102:s
6096:g
6072:.
6069:g
6063:s
6043:,
6040:S
6034:s
6014:S
5994:g
5977:.
5959:2
5955:g
5951:=
5946:1
5942:g
5921:,
5916:1
5912:g
5903:2
5899:g
5876:2
5872:g
5863:1
5859:g
5836:2
5832:g
5809:1
5805:g
5775:,
5772:m
5749:m
5729:,
5726:m
5706:S
5630:.
5627:)
5624:0
5621:,
5618:1
5615:(
5589:2
5584:R
5559:1
5553:x
5547:0
5527:)
5524:y
5521:,
5518:x
5515:(
5489:2
5484:R
5459:,
5455:R
5431:,
5427:R
5396:,
5393:d
5373:c
5353:}
5350:b
5347:,
5344:a
5341:{
5321:.
5318:d
5312:b
5293:,
5290:c
5284:b
5265:,
5262:d
5256:a
5237:,
5234:c
5228:a
5208:}
5205:d
5202:,
5199:c
5196:,
5193:b
5190:,
5187:a
5184:{
5151:.
5134:R
5032:.
5029:P
4987:,
4984:P
4964:S
4939:S
4916:}
4913:4
4910:,
4907:2
4904:,
4901:1
4898:{
4895:=
4892:S
4884:(
4868:S
4819:P
4799:S
4775:P
4750:S
4728:.
4725:g
4705:S
4685:S
4665:g
4645:S
4625:g
4603:.
4600:S
4580:S
4556:S
4529:S
4506:.
4503:P
4497:S
4473:)
4467:,
4464:P
4461:(
4429:)
4423:,
4420:R
4417:(
4397:R
4377:.
4374:)
4368:,
4365:R
4362:(
4342:R
4318:R
4298:R
4278:)
4272:,
4269:R
4266:(
4242:.
4239:R
4233:j
4230:,
4227:i
4204:j
4198:i
4178:R
4136:R
4112:)
4106:,
4103:P
4100:(
4080:.
4077:)
4071:,
4068:P
4065:(
4041:g
4021:)
4015:,
4012:P
4009:(
3989:,
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3890:g
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3864:,
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3832:m
3809:m
3789:,
3786:m
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3711:(
3691:S
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3641:S
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3561:S
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3440:j
3437:=
3434:i
3414:j
3408:i
3388:S
3342:S
3314:.
3311:m
3305:s
3285:m
3265:s
3259:m
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3226:s
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3196:,
3193:P
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3113:)
3107:,
3104:P
3101:(
3081:.
3078:P
3054:)
3048:,
3045:P
3042:(
3022:P
3016:s
2996:,
2993:g
2987:s
2967:)
2961:,
2958:P
2955:(
2935:P
2929:g
2898:x
2878:x
2858:x
2852:x
2824:x
2818:y
2798:y
2792:x
2768:P
2762:y
2759:,
2756:x
2736:)
2730:,
2727:P
2724:(
2704:m
2684:g
2621:.
2618:P
2612:S
2591:)
2585:,
2582:P
2579:(
2557:.
2554:m
2548:s
2528:s
2522:m
2502:S
2496:s
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2446:m
2422:)
2416:,
2413:P
2410:(
2387:.
2384:m
2378:s
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2329:S
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2245:.
2242:P
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2212:)
2206:,
2203:P
2200:(
2169:}
2166:4
2163:,
2160:3
2157:,
2154:2
2151:,
2148:1
2145:{
2142:=
2139:S
2095:P
2074:S
2046:S
2026:S
2002:S
1978:u
1958:S
1952:u
1931:S
1910:S
1889:P
1868:S
1848:u
1824:S
1803:S
1783:g
1763:S
1743:g
1719:,
1716:S
1710:s
1690:u
1684:s
1664:S
1658:u
1638:u
1617:S
1596:S
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1556:,
1553:S
1550:=
1547:P
1527:.
1524:S
1500:S
1480:P
1460:S
1440:P
1420:S
1400:S
1380:.
1377:S
1371:g
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1337:(
1317:S
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1209:.
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1139:S
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1007:S
981:.
978:P
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902:.
899:)
893:,
890:P
887:(
853:)
847:,
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841:(
817:S
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716:,
713:P
710:(
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684:S
678:s
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505:S
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372:.
369:P
363:S
339:)
333:,
330:P
327:(
300:.
297:S
277:S
249:S
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205:S
165:}
159:,
153:,
147:,
144:6
141:,
138:5
135:,
132:3
129:,
126:2
123:,
120:1
117:{
114:=
111:S
91:y
71:x
47:P
20:)
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