22:
2117:
5099:
5079:, 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
6716:
164:
5920:
5590:
5490:
5877:
6456:
5458:
5430:
4505:
2244:
2168:
980:
371:
5134:
6601:
6492:
5960:
5558:
5207:
4915:
1957:
6548:
6416:
6360:
4376:
4079:
901:
7036:
5163:
5008:
4847:
4472:
4428:
4277:
4241:
4157:
4111:
4020:
3941:
3869:
3722:
3596:
3367:
3201:
3112:
3053:
2966:
2735:
2590:
2421:
2211:
1348:
1038:
947:
852:
721:
338:
6998:
6937:
6879:
6071:
5320:
5292:
5264:
5236:
3313:
2995:
2767:
2556:
2386:
2357:
6850:
6182:
6127:
6101:
6042:
5774:
5352:
4203:
3788:
3540:
3514:
3468:
3413:
3264:
3234:
2857:
2823:
2797:
2527:
1718:
1689:
1379:
1141:
1112:
686:
657:
507:
478:
6824:
6798:
6651:
5629:
3831:
3690:
3169:
3021:
2934:
2501:
2451:
2328:
2270:
1663:
1236:
1086:
628:
576:
449:
397:
6743:
5837:
5810:
5526:
2620:
6908:
3442:
1555:
6228:
6515:
6383:
6307:
5728:
5395:
5031:
4986:
4727:
4602:
3988:
3139:
3080:
1526:
1208:
299:
6957:
6763:
6621:
6568:
6327:
6284:
6248:
6202:
6156:
6013:
5993:
5748:
5705:
5372:
4963:
4938:
4867:
4818:
4798:
4774:
4749:
4704:
4684:
4664:
4644:
4624:
4579:
4555:
4528:
4396:
4341:
4317:
4297:
4177:
4135:
4040:
3961:
3909:
3889:
3808:
3762:
3742:
3660:
3640:
3616:
3560:
3488:
3387:
3341:
3284:
2897:
2877:
2703:
2683:
2471:
2296:
2094:
2073:
2045:
2025:
2001:
1977:
1930:
1909:
1888:
1867:
1847:
1823:
1802:
1782:
1762:
1742:
1637:
1616:
1595:
1575:
1499:
1479:
1459:
1439:
1419:
1399:
1316:
1296:
1276:
1256:
1181:
1161:
1060:
1006:
816:
790:
765:
745:
600:
550:
530:
421:
276:
248:
228:
204:
90:
70:
46:
2624:
A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.
2170:
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.
21:
6656:
5641:
7086:
7072:
3141:
This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important
5656:
5404:, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
2899:
is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are
2100:
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative
747:
can have at most one greatest element and it can have at most one least element. Whenever a greatest element of
5460:
the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
95:
5651:
5646:
167:
5055:
The least and greatest element of the whole partially ordered set play a special role and are also called
4917:
in the topmost picture is an example), then the notions of maximal element and greatest element coincide.
5080:
7078:
5882:
5566:
5466:
5842:
4777:
6421:
5438:
5410:
4481:
2220:
2123:
2116:
956:
347:
5117:
6573:
6461:
5925:
5531:
5168:
4876:
2178:
of the set, which are elements that are not strictly smaller than any other element in the set.
1936:
6520:
6388:
6332:
4346:
4049:
871:
7003:
5662:
5146:
4991:
4830:
4445:
4401:
4250:
4211:
4140:
4084:
3993:
3914:
3842:
3695:
3569:
3350:
3174:
3085:
3026:
2939:
2708:
2563:
2394:
2184:
1321:
1011:
985:
920:
825:
694:
311:
255:
6971:
6913:
6855:
6047:
5296:
5268:
5240:
5212:
3289:
2971:
2740:
2532:
2362:
2333:
7110:
7105:
6829:
6161:
6106:
6080:
6018:
5753:
5593:
5325:
4475:
4182:
3767:
3519:
3493:
3447:
3392:
3243:
3210:
2836:
2802:
2776:
2506:
2424:
1694:
1668:
1355:
1117:
1091:
724:
662:
636:
483:
457:
207:
6803:
6777:
6626:
5599:
4945:
If the notions of maximal element and greatest element coincide on every two-element subset
3813:
3669:
3148:
3000:
2913:
2480:
2430:
2307:
2249:
1642:
1215:
1065:
607:
555:
428:
376:
6721:
5815:
5788:
5499:
5076:
5043:
2596:
6884:
3418:
1531:
166:
has one greatest element, viz. 30, and one least element, viz. 1. These elements are also
8:
6207:
6497:
6365:
6289:
5710:
5432:
the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
5377:
5013:
4968:
4709:
4584:
3970:
3121:
3062:
1508:
1190:
281:
6942:
6748:
6606:
6553:
6312:
6269:
6233:
6187:
6141:
5998:
5978:
5733:
5690:
5357:
4948:
4923:
4852:
4803:
4783:
4759:
4734:
4689:
4669:
4649:
4629:
4609:
4564:
4540:
4513:
4381:
4326:
4302:
4282:
4162:
4120:
4025:
3946:
3894:
3874:
3793:
3747:
3727:
3645:
3625:
3601:
3545:
3473:
3372:
3326:
3269:
2882:
2862:
2830:
2688:
2668:
2456:
2281:
2108:(the number 0 in this case) does not imply the existence of a greatest element either.
2079:
2058:
2030:
2010:
1986:
1962:
1915:
1894:
1873:
1852:
1832:
1808:
1787:
1767:
1747:
1727:
1622:
1601:
1580:
1560:
1484:
1464:
1444:
1424:
1404:
1384:
1301:
1281:
1261:
1241:
1166:
1146:
1045:
991:
801:
775:
750:
730:
585:
535:
515:
406:
261:
233:
213:
189:
75:
55:
31:
7082:
7059:
4534:
greatest element. Thus if a set has a greatest element then it is necessarily unique.
2105:
5401:
4821:
4081:
However, the uniqueness conclusion is no longer guaranteed if the preordered set
2275:
2175:
2174:
A greatest element of a subset of a preordered set should not be confused with a
854:
has a greatest element (resp. a least element) then this element is also called
7038:
would have two maximal, but no greatest element, contradicting the coincidence.
3837:
2214:
950:
341:
7099:
5493:
5102:
2653:
2640:
2631:
the maximal element and the greatest element coincide; and it is also called
25:
5670:— a non-strict order such that every non-empty set has a least element
6074:
5963:
5087:
4244:
3542:
holds, which shows that all pairs of distinct (i.e. non-equal) elements in
2904:
2111:
179:
5137:
4870:
4558:
2661:
Role of (in)comparability in distinguishing greatest vs. maximal elements
2628:
2101:
911:
175:
5687:
Of course, in this particular example, there exists only one element in
3598:
can not possibly have a greatest element (because a greatest element of
7058:
The notion of locality requires the function's domain to be at least a
5667:
6881:
In the second case, the definition of maximal element requires that
4751:
has several maximal elements then it cannot have a greatest element.
532:
is on in the above definition, the definition of a least element of
4940:
has a greatest element, the notions coincide, too, as stated above.
2737:
has to do with what elements they are comparable to. Two elements
2907:
partially ordered sets) may have elements that are incomparable.
6765:
and not maximal). This contradicts the ascending chain condition.
5111:
5075:. The notation of 0 and 1 is used preferably when the poset is a
2665:
One of the most important differences between a greatest element
49:
5397:
but no least upper bound, and no greatest element (cf. picture).
4022:
is also partially ordered then it is possible to conclude that
3082:
This is not required of maximal elements. Maximal elements of
5098:
6711:{\displaystyle s_{1}<s_{2}<\cdots <s_{n}<\cdots }
5086:
Further introductory information is found in the article on
3319:
Example where all elements are maximal but none are greatest
1724:
to the definition of a greatest element given before. Thus
4343:
is a greatest element (and thus also a maximal element) of
2104:. This example also demonstrates that the existence of a
5071:(1), respectively. If both exist, the poset is called a
4920:
However, this is not a necessary condition for whenever
2903:
pairs. In general, however, preordered sets (and even
2112:
Contrast to maximal elements and local/absolute maximums
3943:
and moreover, as a consequence of the greatest element
2635:; in the case of function values it is also called the
7006:
6974:
6945:
6916:
6887:
6858:
6832:
6806:
6780:
6751:
6724:
6659:
6653:
Repeating this argument, an infinite ascending chain
6629:
6609:
6576:
6556:
6523:
6500:
6464:
6424:
6391:
6368:
6335:
6315:
6292:
6272:
6236:
6210:
6190:
6164:
6144:
6109:
6083:
6050:
6021:
6001:
5981:
5928:
5885:
5845:
5818:
5791:
5756:
5736:
5713:
5693:
5602:
5569:
5534:
5502:
5469:
5441:
5413:
5380:
5360:
5328:
5299:
5271:
5243:
5215:
5171:
5149:
5120:
5016:
4994:
4971:
4951:
4926:
4879:
4855:
4833:
4806:
4786:
4762:
4737:
4712:
4692:
4672:
4652:
4632:
4612:
4587:
4567:
4543:
4516:
4484:
4448:
4404:
4384:
4349:
4329:
4305:
4285:
4253:
4214:
4185:
4165:
4143:
4123:
4087:
4052:
4028:
3996:
3973:
3949:
3917:
3897:
3877:
3845:
3816:
3796:
3770:
3750:
3730:
3698:
3672:
3648:
3628:
3604:
3572:
3548:
3522:
3496:
3476:
3450:
3421:
3395:
3375:
3353:
3329:
3292:
3272:
3246:
3213:
3177:
3151:
3124:
3088:
3065:
3029:
3003:
2974:
2942:
2916:
2885:
2865:
2839:
2805:
2779:
2743:
2711:
2691:
2671:
2599:
2566:
2535:
2509:
2483:
2459:
2433:
2397:
2365:
2336:
2310:
2284:
2252:
2223:
2187:
2126:
2082:
2061:
2033:
2013:
2003:(however, it may be possible that some other element
1989:
1965:
1939:
1918:
1897:
1876:
1855:
1835:
1811:
1790:
1770:
1750:
1730:
1697:
1671:
1645:
1625:
1604:
1583:
1563:
1534:
1511:
1487:
1467:
1447:
1427:
1407:
1387:
1358:
1324:
1304:
1284:
1264:
1244:
1218:
1193:
1169:
1149:
1120:
1094:
1068:
1048:
1014:
994:
959:
923:
905:
874:
828:
804:
778:
753:
733:
697:
665:
639:
610:
588:
558:
538:
518:
486:
460:
431:
409:
379:
350:
314:
284:
264:
236:
216:
192:
98:
78:
58:
34:
4299:
has exactly one element. All pairs of elements from
4117:also partially ordered. For example, suppose that
3023:; so by its very definition, a greatest element of
2829:if they are not comparable. Because preorders are
2593:is defined to mean a maximal element of the subset
7030:
6992:
6951:
6931:
6902:
6873:
6844:
6818:
6792:
6757:
6737:
6710:
6645:
6615:
6595:
6562:
6542:
6509:
6486:
6450:
6410:
6377:
6354:
6321:
6301:
6278:
6242:
6222:
6196:
6176:
6150:
6121:
6095:
6065:
6036:
6007:
5987:
5954:
5914:
5871:
5831:
5804:
5768:
5742:
5722:
5699:
5623:
5584:
5552:
5520:
5484:
5452:
5424:
5389:
5366:
5346:
5314:
5286:
5258:
5230:
5201:
5157:
5128:
5025:
5002:
4980:
4957:
4932:
4909:
4861:
4841:
4812:
4792:
4768:
4743:
4721:
4698:
4678:
4658:
4638:
4618:
4596:
4573:
4549:
4522:
4499:
4466:
4422:
4390:
4370:
4335:
4311:
4291:
4271:
4235:
4197:
4171:
4151:
4129:
4105:
4073:
4034:
4014:
3982:
3955:
3935:
3903:
3883:
3863:
3825:
3802:
3782:
3756:
3736:
3716:
3684:
3654:
3634:
3610:
3590:
3554:
3534:
3508:
3482:
3462:
3436:
3407:
3381:
3361:
3335:
3307:
3278:
3258:
3228:
3195:
3163:
3133:
3106:
3074:
3047:
3015:
2989:
2960:
2928:
2891:
2871:
2851:
2817:
2791:
2761:
2729:
2697:
2677:
2614:
2584:
2550:
2521:
2495:
2465:
2445:
2415:
2380:
2351:
2322:
2290:
2264:
2238:
2205:
2162:
2088:
2067:
2039:
2019:
1995:
1971:
1951:
1924:
1903:
1882:
1861:
1841:
1817:
1796:
1776:
1756:
1736:
1712:
1683:
1657:
1631:
1610:
1589:
1569:
1549:
1520:
1493:
1473:
1453:
1433:
1413:
1393:
1373:
1342:
1310:
1290:
1270:
1250:
1230:
1202:
1175:
1155:
1135:
1106:
1080:
1054:
1032:
1000:
974:
941:
895:
846:
810:
784:
759:
739:
715:
680:
651:
622:
594:
570:
544:
524:
501:
472:
443:
415:
391:
365:
332:
293:
270:
242:
222:
198:
158:
84:
64:
40:
767:exists and is unique then this element is called
7097:
2657:. Similar conclusions hold for least elements.
2120:In the above divisibility order, the red subset
3618:would, in particular, have to be comparable to
3347:(distinct) elements and define a partial order
7070:
3118:required to be comparable to every element in
7025:
7013:
5341:
5329:
5196:
5172:
4904:
4886:
2157:
2133:
278:that is smaller than every other element of
230:that is greater than every other element of
153:
105:
4686:and moreover, any other maximal element of
4537:If it exists, then the greatest element of
512:By switching the side of the relation that
5046:always has a greatest and a least element.
52:of 60, partially ordered by the relation "
5572:
5472:
5443:
5415:
5154:
5150:
5122:
4999:
4995:
4838:
4834:
4148:
4144:
4137:is a non-empty set and define a preorder
3911:will necessarily be a maximal element of
3358:
3354:
2300:if the following condition is satisfied:
910:Greatest elements are closely related to
5642:Essential supremum and essential infimum
5097:
5036:
3724:because there is exactly one element in
2115:
159:{\displaystyle S=\{1,2,3,5,6,10,15,30\}}
20:
7071:Davey, B. A.; Priestley, H. A. (2002).
3871:does happen to have a greatest element
3145:statement. The defining condition for
2055:for there to exist some upper bound of
1381:In particular, any greatest element of
7098:
3266:(so elements that are incomparable to
3055:must, in particular, be comparable to
2027:). In particular, it is possible for
5750:itself, so the second condition "and
4279:is partially ordered if and only if
552:is obtained. Explicitly, an element
6603:contradicts the incomparability of
6494:The latter must be incomparable to
6362:must exist that is incomparable to
18:Element ≥ (or ≤) each other element
13:
7074:Introduction to Lattices and Order
5596:, this set has upper bounds, e.g.
906:Relationship to upper/lower bounds
170:, respectively, of the red subset.
14:
7122:
5657:Limit superior and limit inferior
5050:
1933:(which can happen if and only if
5915:{\displaystyle g_{2}\leq g_{1},}
5585:{\displaystyle \mathbb {R} ^{2}}
5485:{\displaystyle \mathbb {R} ^{2}}
3662:has no such element). However,
2651:. Together they are called the
6309:but no greatest element. Since
5872:{\displaystyle g_{1}\leq g_{2}}
4398:has at least two elements then
1143:Importantly, an upper bound of
258:, that is, it is an element of
7052:
6962:
6800:be a maximal element, for any
6768:
6451:{\displaystyle s_{1}<s_{2}}
6286:has just one maximal element,
6266:Assume for contradiction that
6253:
6132:
5969:
5779:
5681:
5615:
5603:
5515:
5503:
5114:has no upper bound in the set
4461:
4449:
4417:
4405:
4362:
4350:
4266:
4254:
4100:
4088:
4065:
4053:
4009:
3997:
3930:
3918:
3858:
3846:
3711:
3699:
3585:
3573:
3190:
3178:
3101:
3089:
3042:
3030:
2955:
2943:
2724:
2712:
2579:
2567:
2410:
2398:
2200:
2188:
1891:that is not an upper bound of
1337:
1325:
1027:
1015:
936:
924:
887:
875:
841:
829:
710:
698:
327:
315:
303:
1:
7045:
6718:can be found (such that each
5453:{\displaystyle \mathbb {R} ,}
5425:{\displaystyle \mathbb {R} ,}
4824:, it has one maximal element.
4706:will necessarily be equal to
4666:is also a maximal element of
4500:{\displaystyle S\subseteq P.}
4437:
2239:{\displaystyle S\subseteq P.}
2163:{\displaystyle S=\{1,2,3,4\}}
1528:In the particular case where
1187:required to be an element of
975:{\displaystyle S\subseteq P.}
366:{\displaystyle S\subseteq P.}
6418:cannot be maximal, that is,
5652:Maximal and minimal elements
5647:Initial and terminal objects
5631:It has no least upper bound.
5129:{\displaystyle \mathbb {R} }
3810:itself (which of course, is
2639:, to avoid confusion with a
168:maximal and minimal elements
7:
6596:{\displaystyle s_{2}\leq m}
6487:{\displaystyle s_{2}\in S.}
6158:is a maximal element, then
5995:is the greatest element of
5955:{\displaystyle g_{1}=g_{2}}
5635:
5553:{\displaystyle 0<x<1}
5202:{\displaystyle \{a,b,c,d\}}
5093:
4910:{\displaystyle S=\{1,2,4\}}
4626:is the greatest element of
3744:that is both comparable to
3566:comparable. Consequently,
3171:to be a maximal element of
1952:{\displaystyle u\not \in S}
630:and if it also satisfies:
451:and if it also satisfies:
10:
7127:
7079:Cambridge University Press
6543:{\displaystyle m<s_{2}}
6411:{\displaystyle s_{1}\in S}
6355:{\displaystyle s_{1}\in S}
4581:that is also contained in
4371:{\displaystyle (R,\leq ).}
4074:{\displaystyle (P,\leq ).}
2910:By definition, an element
2473:if and only if there does
1401:is also an upper bound of
896:{\displaystyle (P,\leq ).}
7031:{\displaystyle S=\{a,b\}}
5158:{\displaystyle \,\leq \,}
5003:{\displaystyle \,\leq \,}
4842:{\displaystyle \,\leq \,}
4778:ascending chain condition
4467:{\displaystyle (P,\leq )}
4423:{\displaystyle (R,\leq )}
4272:{\displaystyle (R,\leq )}
4236:{\displaystyle i,j\in R.}
4152:{\displaystyle \,\leq \,}
4106:{\displaystyle (P,\leq )}
4015:{\displaystyle (P,\leq )}
3936:{\displaystyle (P,\leq )}
3864:{\displaystyle (P,\leq )}
3717:{\displaystyle (S,\leq )}
3591:{\displaystyle (S,\leq )}
3362:{\displaystyle \,\leq \,}
3196:{\displaystyle (P,\leq )}
3107:{\displaystyle (P,\leq )}
3048:{\displaystyle (P,\leq )}
2961:{\displaystyle (P,\leq )}
2936:is a greatest element of
2859:is true for all elements
2730:{\displaystyle (P,\leq )}
2585:{\displaystyle (P,\leq )}
2416:{\displaystyle (P,\leq )}
2206:{\displaystyle (P,\leq )}
1983:be a greatest element of
1744:is a greatest element of
1481:is a greatest element of
1343:{\displaystyle (P,\leq )}
1258:is a greatest element of
1033:{\displaystyle (P,\leq )}
942:{\displaystyle (P,\leq )}
847:{\displaystyle (P,\leq )}
716:{\displaystyle (P,\leq )}
333:{\displaystyle (P,\leq )}
210:(poset) is an element of
7000:were incomparable, then
6993:{\displaystyle a,b\in P}
6932:{\displaystyle s\leq m.}
6874:{\displaystyle m\leq s.}
6066:{\displaystyle s\leq g.}
5839:are both greatest, then
5674:
5315:{\displaystyle b\leq d.}
5287:{\displaystyle b\leq c,}
5259:{\displaystyle a\leq d,}
5231:{\displaystyle a\leq c,}
4827:When the restriction of
3692:is a maximal element of
3308:{\displaystyle s\leq m.}
2990:{\displaystyle s\leq g,}
2762:{\displaystyle x,y\in P}
2551:{\displaystyle s\neq m.}
2453:is a maximal element of
2381:{\displaystyle s\leq m.}
2352:{\displaystyle m\leq s,}
2051:have a greatest element
1639:is an element such that
1441:) but an upper bound of
6845:{\displaystyle s\leq m}
6177:{\displaystyle M\leq g}
6122:{\displaystyle g\neq s}
6096:{\displaystyle g\leq s}
6037:{\displaystyle s\in S,}
5769:{\displaystyle \geq m,}
5347:{\displaystyle \{a,b\}}
4820:has a greatest element
4198:{\displaystyle i\leq j}
3783:{\displaystyle \geq m,}
3535:{\displaystyle j\leq i}
3509:{\displaystyle i\leq j}
3463:{\displaystyle i\neq j}
3408:{\displaystyle i\leq j}
3259:{\displaystyle m\leq s}
3229:{\displaystyle s\in P,}
2852:{\displaystyle x\leq x}
2818:{\displaystyle y\leq x}
2792:{\displaystyle x\leq y}
2522:{\displaystyle m\leq s}
1713:{\displaystyle s\in S,}
1684:{\displaystyle s\leq u}
1374:{\displaystyle g\in S.}
1136:{\displaystyle s\in S.}
1107:{\displaystyle s\leq u}
819:is defined similarly.
681:{\displaystyle s\in S.}
652:{\displaystyle l\leq s}
502:{\displaystyle s\in S.}
473:{\displaystyle s\leq g}
7032:
6994:
6959:is a greatest element.
6953:
6933:
6904:
6875:
6846:
6820:
6819:{\displaystyle s\in S}
6794:
6793:{\displaystyle m\in S}
6759:
6739:
6712:
6647:
6646:{\displaystyle s_{1}.}
6617:
6597:
6564:
6544:
6511:
6488:
6452:
6412:
6379:
6356:
6329:is not greatest, some
6323:
6303:
6280:
6244:
6224:
6198:
6178:
6152:
6123:
6097:
6067:
6038:
6009:
5989:
5956:
5916:
5873:
5833:
5806:
5770:
5744:
5724:
5707:that is comparable to
5701:
5663:Upper and lower bounds
5625:
5624:{\displaystyle (1,0).}
5586:
5554:
5522:
5486:
5454:
5426:
5391:
5368:
5348:
5316:
5288:
5260:
5232:
5203:
5159:
5130:
5106:
5027:
5004:
4982:
4959:
4934:
4911:
4863:
4843:
4814:
4794:
4770:
4745:
4723:
4700:
4680:
4660:
4640:
4620:
4598:
4575:
4551:
4524:
4501:
4468:
4424:
4392:
4372:
4337:
4313:
4293:
4273:
4237:
4199:
4173:
4153:
4131:
4107:
4075:
4036:
4016:
3984:
3957:
3937:
3905:
3885:
3865:
3827:
3826:{\displaystyle \leq m}
3804:
3784:
3758:
3738:
3718:
3686:
3685:{\displaystyle m\in S}
3656:
3636:
3612:
3592:
3556:
3536:
3510:
3484:
3464:
3438:
3409:
3383:
3363:
3337:
3309:
3280:
3260:
3230:
3197:
3165:
3164:{\displaystyle m\in P}
3135:
3108:
3076:
3049:
3017:
3016:{\displaystyle s\in P}
2991:
2962:
2930:
2929:{\displaystyle g\in P}
2893:
2873:
2853:
2819:
2793:
2763:
2731:
2699:
2685:and a maximal element
2679:
2643:. The dual terms are
2616:
2586:
2552:
2523:
2497:
2496:{\displaystyle s\in S}
2467:
2447:
2446:{\displaystyle m\in S}
2417:
2382:
2353:
2324:
2323:{\displaystyle s\in S}
2292:
2266:
2265:{\displaystyle m\in S}
2240:
2207:
2171:
2164:
2090:
2069:
2041:
2021:
2007:a greatest element of
1997:
1973:
1953:
1926:
1905:
1884:
1863:
1843:
1819:
1798:
1778:
1758:
1738:
1714:
1685:
1659:
1658:{\displaystyle u\in S}
1633:
1612:
1591:
1571:
1551:
1522:
1495:
1475:
1455:
1435:
1415:
1395:
1375:
1344:
1312:
1292:
1272:
1252:
1232:
1231:{\displaystyle g\in P}
1204:
1177:
1157:
1137:
1108:
1082:
1081:{\displaystyle u\in P}
1056:
1034:
1002:
976:
943:
897:
848:
812:
786:
761:
741:
717:
682:
653:
624:
623:{\displaystyle l\in S}
596:
572:
571:{\displaystyle l\in P}
546:
526:
503:
474:
445:
444:{\displaystyle g\in S}
417:
393:
392:{\displaystyle g\in P}
367:
334:
295:
272:
244:
224:
200:
171:
160:
86:
66:
42:
7033:
6995:
6954:
6934:
6905:
6876:
6847:
6821:
6795:
6760:
6740:
6738:{\displaystyle s_{i}}
6713:
6648:
6618:
6598:
6565:
6545:
6512:
6489:
6453:
6413:
6380:
6357:
6324:
6304:
6281:
6245:
6225:
6199:
6179:
6153:
6124:
6098:
6068:
6039:
6010:
5990:
5957:
5917:
5874:
5834:
5832:{\displaystyle g_{2}}
5807:
5805:{\displaystyle g_{1}}
5771:
5745:
5730:which is necessarily
5725:
5702:
5626:
5594:lexicographical order
5587:
5555:
5523:
5521:{\displaystyle (x,y)}
5487:
5455:
5427:
5392:
5369:
5349:
5317:
5289:
5261:
5233:
5204:
5160:
5131:
5101:
5081:completeness property
5037:Sufficient conditions
5028:
5005:
4983:
4960:
4935:
4912:
4864:
4844:
4815:
4795:
4771:
4746:
4724:
4701:
4681:
4661:
4641:
4621:
4599:
4576:
4552:
4525:
4502:
4476:partially ordered set
4469:
4425:
4393:
4378:So in particular, if
4373:
4338:
4314:
4294:
4274:
4238:
4200:
4174:
4154:
4132:
4108:
4076:
4037:
4017:
3985:
3958:
3938:
3906:
3886:
3866:
3828:
3805:
3785:
3759:
3739:
3719:
3687:
3657:
3637:
3613:
3593:
3557:
3537:
3511:
3485:
3465:
3439:
3410:
3384:
3364:
3338:
3310:
3281:
3261:
3231:
3203:can be reworded as:
3198:
3166:
3136:
3109:
3077:
3050:
3018:
2992:
2963:
2931:
2894:
2874:
2854:
2820:
2794:
2764:
2732:
2700:
2680:
2617:
2615:{\displaystyle S:=P.}
2587:
2553:
2524:
2498:
2468:
2448:
2425:partially ordered set
2418:
2383:
2354:
2325:
2293:
2267:
2241:
2208:
2165:
2119:
2091:
2070:
2042:
2022:
1998:
1974:
1954:
1927:
1906:
1885:
1864:
1849:is an upper bound of
1844:
1820:
1799:
1784:is an upper bound of
1779:
1759:
1739:
1715:
1686:
1660:
1634:
1613:
1592:
1577:is an upper bound of
1572:
1552:
1523:
1496:
1476:
1456:
1436:
1416:
1396:
1376:
1345:
1313:
1298:is an upper bound of
1293:
1273:
1253:
1233:
1205:
1178:
1158:
1138:
1109:
1083:
1057:
1035:
1003:
977:
944:
898:
849:
813:
787:
762:
742:
725:partially ordered set
718:
683:
654:
625:
597:
573:
547:
527:
504:
475:
446:
418:
394:
368:
335:
296:
273:
245:
225:
208:partially ordered set
201:
161:
87:
67:
43:
24:
7004:
6972:
6943:
6914:
6903:{\displaystyle m=s,}
6885:
6856:
6830:
6804:
6778:
6749:
6722:
6657:
6627:
6607:
6574:
6570:'s maximality while
6554:
6521:
6498:
6462:
6422:
6389:
6366:
6333:
6313:
6290:
6270:
6234:
6208:
6188:
6162:
6142:
6107:
6081:
6048:
6019:
5999:
5979:
5926:
5883:
5843:
5816:
5789:
5754:
5734:
5711:
5691:
5600:
5567:
5532:
5500:
5467:
5439:
5411:
5378:
5358:
5326:
5297:
5269:
5241:
5213:
5169:
5147:
5118:
5083:of a partial order.
5077:complemented lattice
5014:
5010:is a total order on
4992:
4969:
4949:
4924:
4877:
4853:
4831:
4804:
4784:
4760:
4735:
4710:
4690:
4670:
4650:
4630:
4610:
4585:
4565:
4541:
4514:
4482:
4446:
4402:
4382:
4347:
4327:
4303:
4283:
4251:
4212:
4183:
4163:
4141:
4121:
4085:
4050:
4026:
3994:
3971:
3963:being comparable to
3947:
3915:
3895:
3875:
3843:
3814:
3794:
3768:
3748:
3728:
3696:
3670:
3646:
3626:
3602:
3570:
3546:
3520:
3494:
3474:
3448:
3437:{\displaystyle i=j.}
3419:
3393:
3373:
3351:
3343:is a set containing
3327:
3290:
3270:
3244:
3211:
3175:
3149:
3122:
3086:
3063:
3027:
3001:
2972:
2940:
2914:
2883:
2863:
2837:
2803:
2777:
2741:
2709:
2705:of a preordered set
2689:
2669:
2597:
2564:
2533:
2507:
2481:
2457:
2431:
2395:
2363:
2334:
2308:
2282:
2250:
2221:
2185:
2124:
2080:
2059:
2031:
2011:
1987:
1963:
1937:
1916:
1895:
1874:
1853:
1833:
1809:
1788:
1768:
1748:
1728:
1722:completely identical
1695:
1669:
1643:
1623:
1602:
1581:
1561:
1550:{\displaystyle P=S,}
1532:
1509:
1485:
1465:
1445:
1425:
1405:
1385:
1356:
1322:
1302:
1282:
1262:
1242:
1216:
1191:
1167:
1147:
1118:
1092:
1066:
1046:
1012:
992:
957:
921:
872:
826:
802:
776:
772:greatest element of
751:
731:
695:
663:
637:
608:
586:
556:
536:
516:
484:
458:
429:
407:
403:greatest element of
377:
348:
312:
282:
262:
234:
214:
190:
96:
76:
56:
32:
6910:so it follows that
6745:is incomparable to
6458:must hold for some
6262:see above. —
6223:{\displaystyle M=g}
6204:is greatest, hence
5560:has no upper bound.
5496:, the set of pairs
4434:greatest elements.
4319:are comparable and
4046:maximal element of
3790:that element being
2629:totally ordered set
2560:maximal element of
1557:the definition of "
7028:
6990:
6949:
6929:
6900:
6871:
6842:
6816:
6790:
6755:
6735:
6708:
6643:
6613:
6593:
6560:
6540:
6510:{\displaystyle m,}
6507:
6484:
6448:
6408:
6378:{\displaystyle m.}
6375:
6352:
6319:
6302:{\displaystyle m,}
6299:
6276:
6240:
6220:
6194:
6174:
6148:
6119:
6093:
6063:
6034:
6005:
5985:
5952:
5912:
5869:
5829:
5802:
5766:
5740:
5723:{\displaystyle m,}
5720:
5697:
5621:
5582:
5550:
5518:
5482:
5450:
5422:
5390:{\displaystyle d,}
5387:
5364:
5344:
5312:
5284:
5256:
5228:
5199:
5155:
5126:
5107:
5026:{\displaystyle P.}
5023:
5000:
4981:{\displaystyle P,}
4978:
4955:
4930:
4907:
4859:
4839:
4810:
4790:
4766:
4741:
4722:{\displaystyle g.}
4719:
4696:
4676:
4656:
4636:
4616:
4597:{\displaystyle S.}
4594:
4571:
4547:
4520:
4497:
4464:
4420:
4388:
4368:
4333:
4309:
4289:
4269:
4233:
4195:
4179:by declaring that
4169:
4149:
4127:
4103:
4071:
4032:
4012:
3983:{\displaystyle P,}
3980:
3953:
3933:
3901:
3881:
3861:
3836:In contrast, if a
3823:
3800:
3780:
3754:
3734:
3714:
3682:
3652:
3632:
3608:
3588:
3552:
3532:
3506:
3480:
3460:
3434:
3405:
3389:by declaring that
3379:
3359:
3333:
3305:
3286:are ignored) then
3276:
3256:
3226:
3193:
3161:
3134:{\displaystyle P.}
3131:
3104:
3075:{\displaystyle P.}
3072:
3045:
3013:
2987:
2958:
2926:
2889:
2869:
2849:
2833:(which means that
2825:; they are called
2815:
2789:
2759:
2727:
2695:
2675:
2612:
2582:
2548:
2519:
2493:
2463:
2443:
2413:
2378:
2349:
2320:
2288:
2262:
2236:
2203:
2172:
2160:
2086:
2065:
2047:to simultaneously
2037:
2017:
1993:
1969:
1949:
1922:
1901:
1880:
1859:
1839:
1815:
1794:
1774:
1754:
1734:
1710:
1681:
1655:
1629:
1608:
1587:
1567:
1547:
1521:{\displaystyle S.}
1518:
1501:if and only if it
1491:
1471:
1451:
1431:
1411:
1391:
1371:
1340:
1308:
1288:
1268:
1248:
1228:
1203:{\displaystyle S.}
1200:
1173:
1153:
1133:
1104:
1078:
1052:
1030:
998:
972:
939:
893:
844:
808:
793:. The terminology
782:
757:
737:
713:
678:
649:
620:
592:
568:
542:
522:
499:
470:
441:
413:
389:
363:
330:
294:{\displaystyle S.}
291:
268:
240:
220:
196:
172:
156:
92:". The red subset
82:
62:
38:
7088:978-0-521-78451-1
7060:topological space
6952:{\displaystyle m}
6758:{\displaystyle m}
6616:{\displaystyle m}
6563:{\displaystyle m}
6322:{\displaystyle m}
6279:{\displaystyle S}
6243:{\displaystyle M}
6197:{\displaystyle g}
6151:{\displaystyle M}
6008:{\displaystyle S}
5988:{\displaystyle g}
5743:{\displaystyle m}
5700:{\displaystyle S}
5367:{\displaystyle c}
5354:has upper bounds
5143:Let the relation
4958:{\displaystyle S}
4933:{\displaystyle S}
4862:{\displaystyle S}
4813:{\displaystyle P}
4793:{\displaystyle S}
4769:{\displaystyle P}
4744:{\displaystyle S}
4699:{\displaystyle S}
4679:{\displaystyle S}
4659:{\displaystyle g}
4639:{\displaystyle S}
4619:{\displaystyle g}
4574:{\displaystyle S}
4550:{\displaystyle S}
4530:can have at most
4523:{\displaystyle S}
4391:{\displaystyle R}
4336:{\displaystyle R}
4312:{\displaystyle R}
4292:{\displaystyle R}
4172:{\displaystyle R}
4130:{\displaystyle R}
4035:{\displaystyle g}
3956:{\displaystyle g}
3904:{\displaystyle g}
3884:{\displaystyle g}
3803:{\displaystyle m}
3757:{\displaystyle m}
3737:{\displaystyle S}
3655:{\displaystyle S}
3635:{\displaystyle S}
3611:{\displaystyle S}
3555:{\displaystyle S}
3483:{\displaystyle S}
3382:{\displaystyle S}
3336:{\displaystyle S}
3279:{\displaystyle m}
2892:{\displaystyle x}
2879:), every element
2872:{\displaystyle x}
2698:{\displaystyle m}
2678:{\displaystyle g}
2466:{\displaystyle S}
2359:then necessarily
2291:{\displaystyle S}
2106:least upper bound
2089:{\displaystyle P}
2068:{\displaystyle S}
2040:{\displaystyle S}
2020:{\displaystyle S}
1996:{\displaystyle S}
1972:{\displaystyle u}
1925:{\displaystyle S}
1904:{\displaystyle S}
1883:{\displaystyle P}
1862:{\displaystyle S}
1842:{\displaystyle u}
1818:{\displaystyle S}
1797:{\displaystyle S}
1777:{\displaystyle g}
1757:{\displaystyle S}
1737:{\displaystyle g}
1632:{\displaystyle u}
1611:{\displaystyle S}
1590:{\displaystyle S}
1570:{\displaystyle u}
1494:{\displaystyle S}
1474:{\displaystyle P}
1454:{\displaystyle S}
1434:{\displaystyle P}
1414:{\displaystyle S}
1394:{\displaystyle S}
1311:{\displaystyle S}
1291:{\displaystyle g}
1271:{\displaystyle S}
1251:{\displaystyle g}
1176:{\displaystyle P}
1156:{\displaystyle S}
1055:{\displaystyle u}
1001:{\displaystyle S}
811:{\displaystyle S}
798:least element of
785:{\displaystyle S}
760:{\displaystyle S}
740:{\displaystyle S}
595:{\displaystyle S}
582:least element of
545:{\displaystyle S}
525:{\displaystyle s}
416:{\displaystyle S}
271:{\displaystyle S}
243:{\displaystyle S}
223:{\displaystyle S}
199:{\displaystyle S}
85:{\displaystyle y}
65:{\displaystyle x}
41:{\displaystyle P}
7118:
7092:
7077:(2nd ed.).
7063:
7056:
7039:
7037:
7035:
7034:
7029:
6999:
6997:
6996:
6991:
6966:
6960:
6958:
6956:
6955:
6950:
6939:In other words,
6938:
6936:
6935:
6930:
6909:
6907:
6906:
6901:
6880:
6878:
6877:
6872:
6851:
6849:
6848:
6843:
6825:
6823:
6822:
6817:
6799:
6797:
6796:
6791:
6772:
6766:
6764:
6762:
6761:
6756:
6744:
6742:
6741:
6736:
6734:
6733:
6717:
6715:
6714:
6709:
6701:
6700:
6682:
6681:
6669:
6668:
6652:
6650:
6649:
6644:
6639:
6638:
6622:
6620:
6619:
6614:
6602:
6600:
6599:
6594:
6586:
6585:
6569:
6567:
6566:
6561:
6549:
6547:
6546:
6541:
6539:
6538:
6516:
6514:
6513:
6508:
6493:
6491:
6490:
6485:
6474:
6473:
6457:
6455:
6454:
6449:
6447:
6446:
6434:
6433:
6417:
6415:
6414:
6409:
6401:
6400:
6384:
6382:
6381:
6376:
6361:
6359:
6358:
6353:
6345:
6344:
6328:
6326:
6325:
6320:
6308:
6306:
6305:
6300:
6285:
6283:
6282:
6277:
6257:
6251:
6249:
6247:
6246:
6241:
6229:
6227:
6226:
6221:
6203:
6201:
6200:
6195:
6183:
6181:
6180:
6175:
6157:
6155:
6154:
6149:
6136:
6130:
6128:
6126:
6125:
6120:
6102:
6100:
6099:
6094:
6077:, this renders (
6072:
6070:
6069:
6064:
6043:
6041:
6040:
6035:
6014:
6012:
6011:
6006:
5994:
5992:
5991:
5986:
5973:
5967:
5961:
5959:
5958:
5953:
5951:
5950:
5938:
5937:
5921:
5919:
5918:
5913:
5908:
5907:
5895:
5894:
5878:
5876:
5875:
5870:
5868:
5867:
5855:
5854:
5838:
5836:
5835:
5830:
5828:
5827:
5811:
5809:
5808:
5803:
5801:
5800:
5783:
5777:
5776:" was redundant.
5775:
5773:
5772:
5767:
5749:
5747:
5746:
5741:
5729:
5727:
5726:
5721:
5706:
5704:
5703:
5698:
5685:
5630:
5628:
5627:
5622:
5591:
5589:
5588:
5583:
5581:
5580:
5575:
5559:
5557:
5556:
5551:
5527:
5525:
5524:
5519:
5491:
5489:
5488:
5483:
5481:
5480:
5475:
5459:
5457:
5456:
5451:
5446:
5431:
5429:
5428:
5423:
5418:
5402:rational numbers
5396:
5394:
5393:
5388:
5373:
5371:
5370:
5365:
5353:
5351:
5350:
5345:
5321:
5319:
5318:
5313:
5293:
5291:
5290:
5285:
5265:
5263:
5262:
5257:
5237:
5235:
5234:
5229:
5208:
5206:
5205:
5200:
5164:
5162:
5161:
5156:
5135:
5133:
5132:
5127:
5125:
5032:
5030:
5029:
5024:
5009:
5007:
5006:
5001:
4987:
4985:
4984:
4979:
4964:
4962:
4961:
4956:
4939:
4937:
4936:
4931:
4916:
4914:
4913:
4908:
4868:
4866:
4865:
4860:
4848:
4846:
4845:
4840:
4819:
4817:
4816:
4811:
4799:
4797:
4796:
4791:
4775:
4773:
4772:
4767:
4750:
4748:
4747:
4742:
4728:
4726:
4725:
4720:
4705:
4703:
4702:
4697:
4685:
4683:
4682:
4677:
4665:
4663:
4662:
4657:
4645:
4643:
4642:
4637:
4625:
4623:
4622:
4617:
4603:
4601:
4600:
4595:
4580:
4578:
4577:
4572:
4556:
4554:
4553:
4548:
4529:
4527:
4526:
4521:
4506:
4504:
4503:
4498:
4473:
4471:
4470:
4465:
4442:Throughout, let
4429:
4427:
4426:
4421:
4397:
4395:
4394:
4389:
4377:
4375:
4374:
4369:
4342:
4340:
4339:
4334:
4318:
4316:
4315:
4310:
4298:
4296:
4295:
4290:
4278:
4276:
4275:
4270:
4242:
4240:
4239:
4234:
4204:
4202:
4201:
4196:
4178:
4176:
4175:
4170:
4158:
4156:
4155:
4150:
4136:
4134:
4133:
4128:
4112:
4110:
4109:
4104:
4080:
4078:
4077:
4072:
4041:
4039:
4038:
4033:
4021:
4019:
4018:
4013:
3989:
3987:
3986:
3981:
3962:
3960:
3959:
3954:
3942:
3940:
3939:
3934:
3910:
3908:
3907:
3902:
3890:
3888:
3887:
3882:
3870:
3868:
3867:
3862:
3832:
3830:
3829:
3824:
3809:
3807:
3806:
3801:
3789:
3787:
3786:
3781:
3763:
3761:
3760:
3755:
3743:
3741:
3740:
3735:
3723:
3721:
3720:
3715:
3691:
3689:
3688:
3683:
3661:
3659:
3658:
3653:
3641:
3639:
3638:
3633:
3617:
3615:
3614:
3609:
3597:
3595:
3594:
3589:
3561:
3559:
3558:
3553:
3541:
3539:
3538:
3533:
3515:
3513:
3512:
3507:
3489:
3487:
3486:
3481:
3469:
3467:
3466:
3461:
3443:
3441:
3440:
3435:
3414:
3412:
3411:
3406:
3388:
3386:
3385:
3380:
3368:
3366:
3365:
3360:
3342:
3340:
3339:
3334:
3314:
3312:
3311:
3306:
3285:
3283:
3282:
3277:
3265:
3263:
3262:
3257:
3235:
3233:
3232:
3227:
3202:
3200:
3199:
3194:
3170:
3168:
3167:
3162:
3140:
3138:
3137:
3132:
3113:
3111:
3110:
3105:
3081:
3079:
3078:
3073:
3054:
3052:
3051:
3046:
3022:
3020:
3019:
3014:
2996:
2994:
2993:
2988:
2967:
2965:
2964:
2959:
2935:
2933:
2932:
2927:
2898:
2896:
2895:
2890:
2878:
2876:
2875:
2870:
2858:
2856:
2855:
2850:
2824:
2822:
2821:
2816:
2798:
2796:
2795:
2790:
2768:
2766:
2765:
2760:
2736:
2734:
2733:
2728:
2704:
2702:
2701:
2696:
2684:
2682:
2681:
2676:
2654:absolute extrema
2649:absolute minimum
2637:absolute maximum
2621:
2619:
2618:
2613:
2591:
2589:
2588:
2583:
2557:
2555:
2554:
2549:
2528:
2526:
2525:
2520:
2502:
2500:
2499:
2494:
2472:
2470:
2469:
2464:
2452:
2450:
2449:
2444:
2422:
2420:
2419:
2414:
2387:
2385:
2384:
2379:
2358:
2356:
2355:
2350:
2329:
2327:
2326:
2321:
2297:
2295:
2294:
2289:
2272:is said to be a
2271:
2269:
2268:
2263:
2245:
2243:
2242:
2237:
2212:
2210:
2209:
2204:
2169:
2167:
2166:
2161:
2095:
2093:
2092:
2087:
2074:
2072:
2071:
2066:
2046:
2044:
2043:
2038:
2026:
2024:
2023:
2018:
2002:
2000:
1999:
1994:
1978:
1976:
1975:
1970:
1958:
1956:
1955:
1950:
1931:
1929:
1928:
1923:
1910:
1908:
1907:
1902:
1889:
1887:
1886:
1881:
1868:
1866:
1865:
1860:
1848:
1846:
1845:
1840:
1824:
1822:
1821:
1816:
1803:
1801:
1800:
1795:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1743:
1741:
1740:
1735:
1719:
1717:
1716:
1711:
1690:
1688:
1687:
1682:
1664:
1662:
1661:
1656:
1638:
1636:
1635:
1630:
1617:
1615:
1614:
1609:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1556:
1554:
1553:
1548:
1527:
1525:
1524:
1519:
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1420:
1418:
1417:
1412:
1400:
1398:
1397:
1392:
1380:
1378:
1377:
1372:
1349:
1347:
1346:
1341:
1317:
1315:
1314:
1309:
1297:
1295:
1294:
1289:
1277:
1275:
1274:
1269:
1257:
1255:
1254:
1249:
1237:
1235:
1234:
1229:
1209:
1207:
1206:
1201:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1142:
1140:
1139:
1134:
1113:
1111:
1110:
1105:
1087:
1085:
1084:
1079:
1061:
1059:
1058:
1053:
1039:
1037:
1036:
1031:
1007:
1005:
1004:
999:
981:
979:
978:
973:
948:
946:
945:
940:
902:
900:
899:
894:
853:
851:
850:
845:
817:
815:
814:
809:
791:
789:
788:
783:
766:
764:
763:
758:
746:
744:
743:
738:
722:
720:
719:
714:
687:
685:
684:
679:
658:
656:
655:
650:
629:
627:
626:
621:
601:
599:
598:
593:
577:
575:
574:
569:
551:
549:
548:
543:
531:
529:
528:
523:
508:
506:
505:
500:
479:
477:
476:
471:
450:
448:
447:
442:
422:
420:
419:
414:
398:
396:
395:
390:
372:
370:
369:
364:
339:
337:
336:
331:
300:
298:
297:
292:
277:
275:
274:
269:
249:
247:
246:
241:
229:
227:
226:
221:
205:
203:
202:
197:
184:greatest element
178:, especially in
165:
163:
162:
157:
91:
89:
88:
83:
71:
69:
68:
63:
47:
45:
44:
39:
7126:
7125:
7121:
7120:
7119:
7117:
7116:
7115:
7096:
7095:
7089:
7067:
7066:
7057:
7053:
7048:
7043:
7042:
7005:
7002:
7001:
6973:
6970:
6969:
6967:
6963:
6944:
6941:
6940:
6915:
6912:
6911:
6886:
6883:
6882:
6857:
6854:
6853:
6831:
6828:
6827:
6805:
6802:
6801:
6779:
6776:
6775:
6773:
6769:
6750:
6747:
6746:
6729:
6725:
6723:
6720:
6719:
6696:
6692:
6677:
6673:
6664:
6660:
6658:
6655:
6654:
6634:
6630:
6628:
6625:
6624:
6608:
6605:
6604:
6581:
6577:
6575:
6572:
6571:
6555:
6552:
6551:
6534:
6530:
6522:
6519:
6518:
6499:
6496:
6495:
6469:
6465:
6463:
6460:
6459:
6442:
6438:
6429:
6425:
6423:
6420:
6419:
6396:
6392:
6390:
6387:
6386:
6367:
6364:
6363:
6340:
6336:
6334:
6331:
6330:
6314:
6311:
6310:
6291:
6288:
6287:
6271:
6268:
6267:
6258:
6254:
6235:
6232:
6231:
6209:
6206:
6205:
6189:
6186:
6185:
6163:
6160:
6159:
6143:
6140:
6139:
6137:
6133:
6108:
6105:
6104:
6082:
6079:
6078:
6049:
6046:
6045:
6020:
6017:
6016:
6000:
5997:
5996:
5980:
5977:
5976:
5974:
5970:
5946:
5942:
5933:
5929:
5927:
5924:
5923:
5903:
5899:
5890:
5886:
5884:
5881:
5880:
5863:
5859:
5850:
5846:
5844:
5841:
5840:
5823:
5819:
5817:
5814:
5813:
5796:
5792:
5790:
5787:
5786:
5784:
5780:
5755:
5752:
5751:
5735:
5732:
5731:
5712:
5709:
5708:
5692:
5689:
5688:
5686:
5682:
5677:
5659:(infimum limit)
5638:
5601:
5598:
5597:
5576:
5571:
5570:
5568:
5565:
5564:
5533:
5530:
5529:
5501:
5498:
5497:
5476:
5471:
5470:
5468:
5465:
5464:
5442:
5440:
5437:
5436:
5414:
5412:
5409:
5408:
5379:
5376:
5375:
5359:
5356:
5355:
5327:
5324:
5323:
5298:
5295:
5294:
5270:
5267:
5266:
5242:
5239:
5238:
5214:
5211:
5210:
5170:
5167:
5166:
5148:
5145:
5144:
5121:
5119:
5116:
5115:
5096:
5053:
5039:
5015:
5012:
5011:
4993:
4990:
4989:
4970:
4967:
4966:
4950:
4947:
4946:
4925:
4922:
4921:
4878:
4875:
4874:
4854:
4851:
4850:
4832:
4829:
4828:
4822:if, and only if
4805:
4802:
4801:
4785:
4782:
4781:
4761:
4758:
4757:
4736:
4733:
4732:
4711:
4708:
4707:
4691:
4688:
4687:
4671:
4668:
4667:
4651:
4648:
4647:
4631:
4628:
4627:
4611:
4608:
4607:
4586:
4583:
4582:
4566:
4563:
4562:
4542:
4539:
4538:
4515:
4512:
4511:
4483:
4480:
4479:
4447:
4444:
4443:
4440:
4403:
4400:
4399:
4383:
4380:
4379:
4348:
4345:
4344:
4328:
4325:
4324:
4304:
4301:
4300:
4284:
4281:
4280:
4252:
4249:
4248:
4247:preordered set
4213:
4210:
4209:
4184:
4181:
4180:
4164:
4161:
4160:
4142:
4139:
4138:
4122:
4119:
4118:
4086:
4083:
4082:
4051:
4048:
4047:
4027:
4024:
4023:
3995:
3992:
3991:
3972:
3969:
3968:
3948:
3945:
3944:
3916:
3913:
3912:
3896:
3893:
3892:
3876:
3873:
3872:
3844:
3841:
3840:
3815:
3812:
3811:
3795:
3792:
3791:
3769:
3766:
3765:
3749:
3746:
3745:
3729:
3726:
3725:
3697:
3694:
3693:
3671:
3668:
3667:
3647:
3644:
3643:
3627:
3624:
3623:
3603:
3600:
3599:
3571:
3568:
3567:
3547:
3544:
3543:
3521:
3518:
3517:
3495:
3492:
3491:
3475:
3472:
3471:
3449:
3446:
3445:
3420:
3417:
3416:
3415:if and only if
3394:
3391:
3390:
3374:
3371:
3370:
3352:
3349:
3348:
3328:
3325:
3324:
3291:
3288:
3287:
3271:
3268:
3267:
3245:
3242:
3241:
3212:
3209:
3208:
3176:
3173:
3172:
3150:
3147:
3146:
3123:
3120:
3119:
3087:
3084:
3083:
3064:
3061:
3060:
3028:
3025:
3024:
3002:
2999:
2998:
2973:
2970:
2969:
2941:
2938:
2937:
2915:
2912:
2911:
2884:
2881:
2880:
2864:
2861:
2860:
2838:
2835:
2834:
2804:
2801:
2800:
2778:
2775:
2774:
2769:are said to be
2742:
2739:
2738:
2710:
2707:
2706:
2690:
2687:
2686:
2670:
2667:
2666:
2598:
2595:
2594:
2565:
2562:
2561:
2534:
2531:
2530:
2508:
2505:
2504:
2482:
2479:
2478:
2458:
2455:
2454:
2432:
2429:
2428:
2396:
2393:
2392:
2364:
2361:
2360:
2335:
2332:
2331:
2309:
2306:
2305:
2283:
2280:
2279:
2276:maximal element
2251:
2248:
2247:
2222:
2219:
2218:
2186:
2183:
2182:
2176:maximal element
2125:
2122:
2121:
2114:
2081:
2078:
2077:
2060:
2057:
2056:
2032:
2029:
2028:
2012:
2009:
2008:
1988:
1985:
1984:
1964:
1961:
1960:
1938:
1935:
1934:
1917:
1914:
1913:
1896:
1893:
1892:
1875:
1872:
1871:
1854:
1851:
1850:
1834:
1831:
1830:
1810:
1807:
1806:
1789:
1786:
1785:
1769:
1766:
1765:
1764:if and only if
1749:
1746:
1745:
1729:
1726:
1725:
1696:
1693:
1692:
1670:
1667:
1666:
1644:
1641:
1640:
1624:
1621:
1620:
1603:
1600:
1599:
1582:
1579:
1578:
1562:
1559:
1558:
1533:
1530:
1529:
1510:
1507:
1506:
1486:
1483:
1482:
1466:
1463:
1462:
1446:
1443:
1442:
1426:
1423:
1422:
1406:
1403:
1402:
1386:
1383:
1382:
1357:
1354:
1353:
1323:
1320:
1319:
1303:
1300:
1299:
1283:
1280:
1279:
1278:if and only if
1263:
1260:
1259:
1243:
1240:
1239:
1217:
1214:
1213:
1192:
1189:
1188:
1168:
1165:
1164:
1148:
1145:
1144:
1119:
1116:
1115:
1093:
1090:
1089:
1067:
1064:
1063:
1047:
1044:
1043:
1013:
1010:
1009:
993:
990:
989:
958:
955:
954:
922:
919:
918:
908:
873:
870:
869:
827:
824:
823:
803:
800:
799:
777:
774:
773:
752:
749:
748:
732:
729:
728:
696:
693:
692:
664:
661:
660:
638:
635:
634:
609:
606:
605:
587:
584:
583:
557:
554:
553:
537:
534:
533:
517:
514:
513:
485:
482:
481:
459:
456:
455:
430:
427:
426:
408:
405:
404:
378:
375:
374:
349:
346:
345:
313:
310:
309:
306:
283:
280:
279:
263:
260:
259:
235:
232:
231:
215:
212:
211:
191:
188:
187:
97:
94:
93:
77:
74:
73:
57:
54:
53:
33:
30:
29:
19:
12:
11:
5:
7124:
7114:
7113:
7108:
7094:
7093:
7087:
7065:
7064:
7050:
7049:
7047:
7044:
7041:
7040:
7027:
7024:
7021:
7018:
7015:
7012:
7009:
6989:
6986:
6983:
6980:
6977:
6961:
6948:
6928:
6925:
6922:
6919:
6899:
6896:
6893:
6890:
6870:
6867:
6864:
6861:
6841:
6838:
6835:
6815:
6812:
6809:
6789:
6786:
6783:
6767:
6754:
6732:
6728:
6707:
6704:
6699:
6695:
6691:
6688:
6685:
6680:
6676:
6672:
6667:
6663:
6642:
6637:
6633:
6612:
6592:
6589:
6584:
6580:
6559:
6537:
6533:
6529:
6526:
6506:
6503:
6483:
6480:
6477:
6472:
6468:
6445:
6441:
6437:
6432:
6428:
6407:
6404:
6399:
6395:
6374:
6371:
6351:
6348:
6343:
6339:
6318:
6298:
6295:
6275:
6252:
6239:
6219:
6216:
6213:
6193:
6173:
6170:
6167:
6147:
6131:
6118:
6115:
6112:
6092:
6089:
6086:
6062:
6059:
6056:
6053:
6033:
6030:
6027:
6024:
6004:
5984:
5968:
5949:
5945:
5941:
5936:
5932:
5911:
5906:
5902:
5898:
5893:
5889:
5866:
5862:
5858:
5853:
5849:
5826:
5822:
5799:
5795:
5778:
5765:
5762:
5759:
5739:
5719:
5716:
5696:
5679:
5678:
5676:
5673:
5672:
5671:
5665:
5660:
5654:
5649:
5644:
5637:
5634:
5633:
5632:
5620:
5617:
5614:
5611:
5608:
5605:
5579:
5574:
5561:
5549:
5546:
5543:
5540:
5537:
5517:
5514:
5511:
5508:
5505:
5479:
5474:
5461:
5449:
5445:
5433:
5421:
5417:
5405:
5398:
5386:
5383:
5363:
5343:
5340:
5337:
5334:
5331:
5311:
5308:
5305:
5302:
5283:
5280:
5277:
5274:
5255:
5252:
5249:
5246:
5227:
5224:
5221:
5218:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5153:
5141:
5124:
5110:The subset of
5095:
5092:
5052:
5051:Top and bottom
5049:
5048:
5047:
5038:
5035:
5034:
5033:
5022:
5019:
4998:
4977:
4974:
4954:
4943:
4942:
4941:
4929:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4858:
4837:
4825:
4809:
4789:
4776:satisfies the
4765:
4754:
4753:
4752:
4740:
4731:Thus if a set
4718:
4715:
4695:
4675:
4655:
4635:
4615:
4604:
4593:
4590:
4570:
4546:
4535:
4533:
4519:
4496:
4493:
4490:
4487:
4463:
4460:
4457:
4454:
4451:
4439:
4436:
4433:
4419:
4416:
4413:
4410:
4407:
4387:
4367:
4364:
4361:
4358:
4355:
4352:
4332:
4322:
4308:
4288:
4268:
4265:
4262:
4259:
4256:
4232:
4229:
4226:
4223:
4220:
4217:
4208:holds for all
4207:
4194:
4191:
4188:
4168:
4147:
4126:
4116:
4102:
4099:
4096:
4093:
4090:
4070:
4067:
4064:
4061:
4058:
4055:
4045:
4031:
4011:
4008:
4005:
4002:
3999:
3979:
3976:
3966:
3952:
3932:
3929:
3926:
3923:
3920:
3900:
3880:
3860:
3857:
3854:
3851:
3848:
3838:preordered set
3822:
3819:
3799:
3779:
3776:
3773:
3753:
3733:
3713:
3710:
3707:
3704:
3701:
3681:
3678:
3675:
3665:
3651:
3631:
3621:
3607:
3587:
3584:
3581:
3578:
3575:
3565:
3551:
3531:
3528:
3525:
3505:
3502:
3499:
3479:
3459:
3456:
3453:
3433:
3430:
3427:
3424:
3404:
3401:
3398:
3378:
3357:
3346:
3332:
3321:
3320:
3316:
3315:
3304:
3301:
3298:
3295:
3275:
3255:
3252:
3249:
3239:
3225:
3222:
3219:
3216:
3192:
3189:
3186:
3183:
3180:
3160:
3157:
3154:
3144:
3130:
3127:
3117:
3103:
3100:
3097:
3094:
3091:
3071:
3068:
3058:
3044:
3041:
3038:
3035:
3032:
3012:
3009:
3006:
2986:
2983:
2980:
2977:
2957:
2954:
2951:
2948:
2945:
2925:
2922:
2919:
2902:
2888:
2868:
2848:
2845:
2842:
2828:
2814:
2811:
2808:
2788:
2785:
2782:
2772:
2758:
2755:
2752:
2749:
2746:
2726:
2723:
2720:
2717:
2714:
2694:
2674:
2663:
2662:
2611:
2608:
2605:
2602:
2592:
2581:
2578:
2575:
2572:
2569:
2547:
2544:
2541:
2538:
2518:
2515:
2512:
2492:
2489:
2486:
2476:
2462:
2442:
2439:
2436:
2412:
2409:
2406:
2403:
2400:
2389:
2388:
2377:
2374:
2371:
2368:
2348:
2345:
2342:
2339:
2319:
2316:
2313:
2298:
2287:
2261:
2258:
2255:
2235:
2232:
2229:
2226:
2215:preordered set
2202:
2199:
2196:
2193:
2190:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2113:
2110:
2096:
2085:
2064:
2054:
2050:
2036:
2016:
2006:
1992:
1982:
1968:
1948:
1945:
1942:
1932:
1921:
1900:
1890:
1879:
1858:
1838:
1825:
1814:
1793:
1773:
1753:
1733:
1723:
1709:
1706:
1703:
1700:
1680:
1677:
1674:
1654:
1651:
1648:
1628:
1618:
1607:
1586:
1566:
1546:
1543:
1540:
1537:
1517:
1514:
1504:
1490:
1470:
1450:
1430:
1410:
1390:
1370:
1367:
1364:
1361:
1352:
1339:
1336:
1333:
1330:
1327:
1307:
1287:
1267:
1247:
1227:
1224:
1221:
1199:
1196:
1186:
1172:
1152:
1132:
1129:
1126:
1123:
1103:
1100:
1097:
1077:
1074:
1071:
1051:
1042:is an element
1040:
1029:
1026:
1023:
1020:
1017:
997:
971:
968:
965:
962:
951:preordered set
938:
935:
932:
929:
926:
907:
904:
892:
889:
886:
883:
880:
877:
867:
860:
843:
840:
837:
834:
831:
807:
797:
781:
771:
756:
736:
712:
709:
706:
703:
700:
689:
688:
677:
674:
671:
668:
648:
645:
642:
619:
616:
613:
603:
591:
578:is said to be
567:
564:
561:
541:
521:
510:
509:
498:
495:
492:
489:
469:
466:
463:
440:
437:
434:
424:
412:
399:is said to be
388:
385:
382:
362:
359:
356:
353:
342:preordered set
329:
326:
323:
320:
317:
305:
302:
290:
287:
267:
239:
219:
195:
155:
152:
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
81:
61:
37:
17:
9:
6:
4:
3:
2:
7123:
7112:
7109:
7107:
7104:
7103:
7101:
7090:
7084:
7080:
7076:
7075:
7069:
7068:
7061:
7055:
7051:
7022:
7019:
7016:
7010:
7007:
6987:
6984:
6981:
6978:
6975:
6965:
6946:
6926:
6923:
6920:
6917:
6897:
6894:
6891:
6888:
6868:
6865:
6862:
6859:
6839:
6836:
6833:
6813:
6810:
6807:
6787:
6784:
6781:
6771:
6752:
6730:
6726:
6705:
6702:
6697:
6693:
6689:
6686:
6683:
6678:
6674:
6670:
6665:
6661:
6640:
6635:
6631:
6610:
6590:
6587:
6582:
6578:
6557:
6535:
6531:
6527:
6524:
6504:
6501:
6481:
6478:
6475:
6470:
6466:
6443:
6439:
6435:
6430:
6426:
6405:
6402:
6397:
6393:
6372:
6369:
6349:
6346:
6341:
6337:
6316:
6296:
6293:
6273:
6265:
6261:
6256:
6237:
6217:
6214:
6211:
6191:
6171:
6168:
6165:
6145:
6135:
6129:) impossible.
6116:
6113:
6110:
6090:
6087:
6084:
6076:
6060:
6057:
6054:
6051:
6031:
6028:
6025:
6022:
6002:
5982:
5972:
5965:
5947:
5943:
5939:
5934:
5930:
5909:
5904:
5900:
5896:
5891:
5887:
5864:
5860:
5856:
5851:
5847:
5824:
5820:
5797:
5793:
5782:
5763:
5760:
5757:
5737:
5717:
5714:
5694:
5684:
5680:
5669:
5666:
5664:
5661:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5639:
5618:
5612:
5609:
5606:
5595:
5577:
5562:
5547:
5544:
5541:
5538:
5535:
5512:
5509:
5506:
5495:
5494:product order
5477:
5462:
5447:
5434:
5419:
5406:
5403:
5399:
5384:
5381:
5361:
5338:
5335:
5332:
5309:
5306:
5303:
5300:
5281:
5278:
5275:
5272:
5253:
5250:
5247:
5244:
5225:
5222:
5219:
5216:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5151:
5142:
5139:
5113:
5109:
5108:
5104:
5103:Hasse diagram
5100:
5091:
5089:
5084:
5082:
5078:
5074:
5073:bounded poset
5070:
5066:
5062:
5058:
5045:
5041:
5040:
5020:
5017:
4996:
4975:
4972:
4952:
4944:
4927:
4919:
4918:
4901:
4898:
4895:
4892:
4889:
4883:
4880:
4872:
4856:
4835:
4826:
4823:
4807:
4787:
4779:
4763:
4755:
4738:
4730:
4729:
4716:
4713:
4693:
4673:
4653:
4633:
4613:
4605:
4591:
4588:
4568:
4560:
4544:
4536:
4531:
4517:
4509:
4508:
4507:
4494:
4491:
4488:
4485:
4477:
4458:
4455:
4452:
4435:
4431:
4430:has multiple
4414:
4411:
4408:
4385:
4365:
4359:
4356:
4353:
4330:
4320:
4306:
4286:
4263:
4260:
4257:
4246:
4230:
4227:
4224:
4221:
4218:
4215:
4205:
4192:
4189:
4186:
4166:
4145:
4124:
4114:
4097:
4094:
4091:
4068:
4062:
4059:
4056:
4043:
4029:
4006:
4003:
4000:
3977:
3974:
3964:
3950:
3927:
3924:
3921:
3898:
3878:
3855:
3852:
3849:
3839:
3834:
3820:
3817:
3797:
3777:
3774:
3771:
3751:
3731:
3708:
3705:
3702:
3679:
3676:
3673:
3663:
3649:
3629:
3619:
3605:
3582:
3579:
3576:
3563:
3549:
3529:
3526:
3523:
3503:
3500:
3497:
3490:then neither
3477:
3457:
3454:
3451:
3431:
3428:
3425:
3422:
3402:
3399:
3396:
3376:
3355:
3344:
3330:
3323:Suppose that
3318:
3317:
3302:
3299:
3296:
3293:
3273:
3253:
3250:
3247:
3240:
3237:
3223:
3220:
3217:
3214:
3206:
3205:
3204:
3187:
3184:
3181:
3158:
3155:
3152:
3142:
3128:
3125:
3115:
3098:
3095:
3092:
3069:
3066:
3056:
3039:
3036:
3033:
3010:
3007:
3004:
2984:
2981:
2978:
2975:
2952:
2949:
2946:
2923:
2920:
2917:
2908:
2906:
2900:
2886:
2866:
2846:
2843:
2840:
2832:
2826:
2812:
2809:
2806:
2786:
2783:
2780:
2770:
2756:
2753:
2750:
2747:
2744:
2721:
2718:
2715:
2692:
2672:
2660:
2659:
2658:
2656:
2655:
2650:
2646:
2642:
2641:local maximum
2638:
2634:
2630:
2625:
2622:
2609:
2606:
2603:
2600:
2576:
2573:
2570:
2559:
2545:
2542:
2539:
2536:
2516:
2513:
2510:
2490:
2487:
2484:
2474:
2460:
2440:
2437:
2434:
2426:
2407:
2404:
2401:
2375:
2372:
2369:
2366:
2346:
2343:
2340:
2337:
2317:
2314:
2311:
2303:
2302:
2301:
2299:
2285:
2277:
2274:
2259:
2256:
2253:
2233:
2230:
2227:
2224:
2216:
2197:
2194:
2191:
2179:
2177:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2130:
2127:
2118:
2109:
2107:
2103:
2098:
2083:
2075:
2062:
2052:
2048:
2034:
2014:
2004:
1990:
1980:
1966:
1946:
1943:
1940:
1919:
1911:
1898:
1877:
1869:
1856:
1836:
1827:
1812:
1804:
1791:
1771:
1751:
1731:
1721:
1707:
1704:
1701:
1698:
1678:
1675:
1672:
1652:
1649:
1646:
1626:
1605:
1597:
1584:
1564:
1544:
1541:
1538:
1535:
1515:
1512:
1502:
1488:
1468:
1448:
1428:
1408:
1388:
1368:
1365:
1362:
1359:
1350:
1334:
1331:
1328:
1305:
1285:
1265:
1245:
1225:
1222:
1219:
1210:
1197:
1194:
1184:
1170:
1150:
1130:
1127:
1124:
1121:
1101:
1098:
1095:
1075:
1072:
1069:
1049:
1041:
1024:
1021:
1018:
995:
987:
984:
969:
966:
963:
960:
952:
933:
930:
927:
915:
913:
903:
890:
884:
881:
878:
866:
862:
859:
855:
838:
835:
832:
820:
818:
805:
795:
792:
779:
769:
754:
734:
726:
707:
704:
701:
675:
672:
669:
666:
646:
643:
640:
633:
632:
631:
617:
614:
611:
602:
589:
579:
565:
562:
559:
539:
519:
496:
493:
490:
487:
467:
464:
461:
454:
453:
452:
438:
435:
432:
423:
410:
400:
386:
383:
380:
360:
357:
354:
351:
343:
324:
321:
318:
301:
288:
285:
265:
257:
253:
252:least element
237:
217:
209:
193:
185:
181:
177:
169:
150:
147:
144:
141:
138:
135:
132:
129:
126:
123:
120:
117:
114:
111:
108:
102:
99:
79:
59:
51:
35:
27:
26:Hasse diagram
23:
16:
7111:Superlatives
7106:Order theory
7073:
7054:
6964:
6770:
6550:contradicts
6263:
6259:
6255:
6134:
6075:antisymmetry
5971:
5964:antisymmetry
5781:
5683:
5209:be given by
5138:real numbers
5105:of example 2
5088:order theory
5085:
5072:
5068:
5064:
5060:
5056:
5054:
4441:
3835:
3345:at least two
3322:
3236:
2909:
2827:incomparable
2664:
2652:
2648:
2644:
2636:
2632:
2626:
2623:
2390:
2273:
2180:
2173:
2102:real numbers
2099:
1828:
1211:
983:
916:
912:upper bounds
909:
864:
857:
821:
794:
768:
690:
581:
511:
402:
307:
251:
186:of a subset
183:
180:order theory
173:
15:
6517:too, since
6250:is maximal.
4871:total order
4780:, a subset
4559:upper bound
4323:element of
3967:element of
3622:element of
3059:element in
2246:An element
1619:" becomes:
986:upper bound
373:An element
304:Definitions
254:is defined
250:. The term
176:mathematics
28:of the set
7100:Categories
7046:References
5922:and hence
5668:Well-order
4438:Properties
3470:belong to
2997:for every
2771:comparable
2503:such that
2477:exist any
2330:satisfies
1062:such that
723:is also a
6985:∈
6921:≤
6863:≤
6837:≤
6811:∈
6785:∈
6706:⋯
6687:⋯
6588:≤
6476:∈
6403:∈
6347:∈
6169:≤
6114:≠
6088:≤
6055:≤
6026:∈
5897:≤
5857:≤
5758:≥
5592:with the
5492:with the
5304:≤
5276:≤
5248:≤
5220:≤
5152:≤
5042:A finite
4997:≤
4836:≤
4489:⊆
4459:≤
4415:≤
4360:≤
4264:≤
4225:∈
4190:≤
4146:≤
4098:≤
4063:≤
4007:≤
3928:≤
3856:≤
3818:≤
3772:≥
3709:≤
3677:∈
3583:≤
3527:≤
3501:≤
3455:≠
3400:≤
3356:≤
3297:≤
3251:≤
3218:∈
3188:≤
3156:∈
3099:≤
3040:≤
3008:∈
2979:≤
2953:≤
2921:∈
2844:≤
2831:reflexive
2810:≤
2784:≤
2754:∈
2722:≤
2577:≤
2540:≠
2514:≤
2488:∈
2438:∈
2408:≤
2370:≤
2341:≤
2315:∈
2304:whenever
2257:∈
2228:⊆
2198:≤
1720:which is
1702:∈
1676:≤
1650:∈
1363:∈
1335:≤
1223:∈
1125:∈
1099:≤
1073:∈
1025:≤
964:⊆
934:≤
885:≤
839:≤
708:≤
670:∈
644:≤
615:∈
563:∈
491:∈
465:≤
436:∈
384:∈
355:⊆
325:≤
6260:Only if:
5636:See also
5322:The set
5112:integers
5094:Examples
5067:(0) and
5063:(⊤), or
5059:(⊥) and
4478:and let
4432:distinct
4245:directed
3666:element
3207:For all
2905:directed
2901:distinct
2217:and let
1944:∉
1691:for all
1114:for all
953:and let
659:for all
480:for all
344:and let
72:divides
50:divisors
6826:either
5400:In the
4042:is the
2645:minimum
2633:maximum
1959:) then
1503:belongs
861:(resp.
7085:
6385:Hence
6230:since
6184:since
5057:bottom
4557:is an
4510:A set
4206:always
865:bottom
256:dually
182:, the
6044:then
5675:Notes
5528:with
5044:chain
4988:then
4869:is a
4646:then
4474:be a
4321:every
3965:every
3891:then
3664:every
3620:every
3057:every
2627:In a
2427:then
2423:is a
2213:be a
1238:then
1088:and
949:be a
868:) of
727:then
340:be a
206:of a
7083:ISBN
6774:Let
6703:<
6690:<
6684:<
6671:<
6623:and
6528:<
6436:<
6103:and
6015:and
5879:and
5812:and
5545:<
5539:<
5374:and
5069:unit
5065:zero
4243:The
4044:only
3764:and
3642:but
3562:are
3516:nor
3114:are
2647:and
2529:and
2181:Let
1979:can
1665:and
1421:(in
917:Let
308:Let
6968:If
6852:or
6264:If:
6138:If
6073:By
5975:If
5962:by
5785:If
5563:In
5463:In
5435:In
5407:In
5165:on
5136:of
5061:top
4965:of
4849:to
4800:of
4756:If
4606:If
4561:of
4532:one
4159:on
4115:not
4113:is
3990:if
3833:).
3444:If
3369:on
3116:not
2968:if
2799:or
2773:if
2475:not
2391:If
2278:of
2097:.
2076:in
2053:and
2049:not
1981:not
1912:in
1870:in
1829:If
1826:.
1805:in
1598:in
1505:to
1461:in
1351:and
1318:in
1212:If
1185:not
1183:is
1163:in
1008:in
988:of
982:An
914:.
858:top
822:If
796:the
770:the
691:If
604:if
425:if
174:In
48:of
7102::
7081:.
5090:.
3564:in
3238:IF
3143:if
2604::=
2558:A
2005:is
863:a
856:a
580:a
401:a
151:30
145:15
139:10
7091:.
7062:.
7026:}
7023:b
7020:,
7017:a
7014:{
7011:=
7008:S
6988:P
6982:b
6979:,
6976:a
6947:m
6927:.
6924:m
6918:s
6898:,
6895:s
6892:=
6889:m
6869:.
6866:s
6860:m
6840:m
6834:s
6814:S
6808:s
6788:S
6782:m
6753:m
6731:i
6727:s
6698:n
6694:s
6679:2
6675:s
6666:1
6662:s
6641:.
6636:1
6632:s
6611:m
6591:m
6583:2
6579:s
6558:m
6536:2
6532:s
6525:m
6505:,
6502:m
6482:.
6479:S
6471:2
6467:s
6444:2
6440:s
6431:1
6427:s
6406:S
6398:1
6394:s
6373:.
6370:m
6350:S
6342:1
6338:s
6317:m
6297:,
6294:m
6274:S
6238:M
6218:g
6215:=
6212:M
6192:g
6172:g
6166:M
6146:M
6117:s
6111:g
6091:s
6085:g
6061:.
6058:g
6052:s
6032:,
6029:S
6023:s
6003:S
5983:g
5966:.
5948:2
5944:g
5940:=
5935:1
5931:g
5910:,
5905:1
5901:g
5892:2
5888:g
5865:2
5861:g
5852:1
5848:g
5825:2
5821:g
5798:1
5794:g
5764:,
5761:m
5738:m
5718:,
5715:m
5695:S
5619:.
5616:)
5613:0
5610:,
5607:1
5604:(
5578:2
5573:R
5548:1
5542:x
5536:0
5516:)
5513:y
5510:,
5507:x
5504:(
5478:2
5473:R
5448:,
5444:R
5420:,
5416:R
5385:,
5382:d
5362:c
5342:}
5339:b
5336:,
5333:a
5330:{
5310:.
5307:d
5301:b
5282:,
5279:c
5273:b
5254:,
5251:d
5245:a
5226:,
5223:c
5217:a
5197:}
5194:d
5191:,
5188:c
5185:,
5182:b
5179:,
5176:a
5173:{
5140:.
5123:R
5021:.
5018:P
4976:,
4973:P
4953:S
4928:S
4905:}
4902:4
4899:,
4896:2
4893:,
4890:1
4887:{
4884:=
4881:S
4873:(
4857:S
4808:P
4788:S
4764:P
4739:S
4717:.
4714:g
4694:S
4674:S
4654:g
4634:S
4614:g
4592:.
4589:S
4569:S
4545:S
4518:S
4495:.
4492:P
4486:S
4462:)
4456:,
4453:P
4450:(
4418:)
4412:,
4409:R
4406:(
4386:R
4366:.
4363:)
4357:,
4354:R
4351:(
4331:R
4307:R
4287:R
4267:)
4261:,
4258:R
4255:(
4231:.
4228:R
4222:j
4219:,
4216:i
4193:j
4187:i
4167:R
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4101:)
4095:,
4092:P
4089:(
4069:.
4066:)
4060:,
4057:P
4054:(
4030:g
4010:)
4004:,
4001:P
3998:(
3978:,
3975:P
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3847:(
3821:m
3798:m
3778:,
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3700:(
3680:S
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3606:S
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3550:S
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3504:j
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3429:j
3426:=
3423:i
3403:j
3397:i
3377:S
3331:S
3303:.
3300:m
3294:s
3274:m
3254:s
3248:m
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3185:,
3182:P
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3126:P
3102:)
3096:,
3093:P
3090:(
3070:.
3067:P
3043:)
3037:,
3034:P
3031:(
3011:P
3005:s
2985:,
2982:g
2976:s
2956:)
2950:,
2947:P
2944:(
2924:P
2918:g
2887:x
2867:x
2847:x
2841:x
2813:x
2807:y
2787:y
2781:x
2757:P
2751:y
2748:,
2745:x
2725:)
2719:,
2716:P
2713:(
2693:m
2673:g
2610:.
2607:P
2601:S
2580:)
2574:,
2571:P
2568:(
2546:.
2543:m
2537:s
2517:s
2511:m
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2485:s
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2435:m
2411:)
2405:,
2402:P
2399:(
2376:.
2373:m
2367:s
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2254:m
2234:.
2231:P
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2201:)
2195:,
2192:P
2189:(
2158:}
2155:4
2152:,
2149:3
2146:,
2143:2
2140:,
2137:1
2134:{
2131:=
2128:S
2084:P
2063:S
2035:S
2015:S
1991:S
1967:u
1947:S
1941:u
1920:S
1899:S
1878:P
1857:S
1837:u
1813:S
1792:S
1772:g
1752:S
1732:g
1708:,
1705:S
1699:s
1679:u
1673:s
1653:S
1647:u
1627:u
1606:S
1585:S
1565:u
1545:,
1542:S
1539:=
1536:P
1516:.
1513:S
1489:S
1469:P
1449:S
1429:P
1409:S
1389:S
1369:.
1366:S
1360:g
1338:)
1332:,
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1326:(
1306:S
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1198:.
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1131:.
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1016:(
996:S
970:.
967:P
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928:P
925:(
891:.
888:)
882:,
879:P
876:(
842:)
836:,
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830:(
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705:,
702:P
699:(
676:.
673:S
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641:l
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612:l
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494:S
488:s
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361:.
358:P
352:S
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322:,
319:P
316:(
289:.
286:S
266:S
238:S
218:S
194:S
154:}
148:,
142:,
136:,
133:6
130:,
127:5
124:,
121:3
118:,
115:2
112:,
109:1
106:{
103:=
100:S
80:y
60:x
36:P
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