583:
568:
20:
1889:. In fact, every semiorder is quasitransitive, and quasitransitivity is invariant to adding all pairs of incomparable items. Removing all non-vertical red lines from the topmost image results in a Hasse diagram for a relation that is still quasitransitive, but violates both axiom 2 and 3; this relation might no longer be useful as a preference ordering.
1675:. Not every partial order leads to a semiorder in this way, however: The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, but the others do not. A partial order that includes four elements forming two two-element chains would lead to a relation
1720:
Every strict weak ordering < is also a semi-order. More particularly, transitivity of < and transitivity of incomparability with respect to < together imply the above axiom 2, while transitivity of incomparability alone implies axiom 3. The semiorder shown in the top image is not a strict
1058:
Conversely, every finite partial order that avoids the two forbidden four-point orderings described above can be given utility values making it into a semiorder. Therefore, rather than being a consequence of a definition in terms of utility, these forbidden orderings, or equivalent systems of
2239:, Theorem 3 describes a more general situation in which the threshold for comparability between two utilities is a function of the utility rather than being identically 1; however, this does not lead to a different class of orderings.
1146:
For orderings on infinite sets of elements, the orderings that can be defined by utility functions and the orderings that can be defined by forbidden four-point orders differ from each other. For instance, if a semiorder
1707:
that violates the second semiorder axiom, and a partial order that includes four elements forming a three-element chain and an unrelated item would violate the third semiorder axiom (cf. pictures in section
2191:
steps from the first semiorder to the second one, in such a way that each step of the path adds or removes a single order relation and each intermediate state in the path is itself a semiorder.
1083:
elements is given only in terms of the order relation between its pairs of elements, obeying these axioms, then it is possible to construct a utility function that represents the order in time
1983:
1863:
516:
966:
896:
767:
697:
422:. Set a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each other are declared incomparable, and define a binary relation
832:
1595:
1560:
1236:
1705:
1268:
1177:
1117:
548:
1673:
1647:
1528:
1502:
1476:
1424:
1398:
1372:
1294:
1621:
1320:
992:
922:
793:
723:
653:
627:
466:
440:
1450:
1346:
2189:
2169:
2145:
2122:
2102:
2082:
2062:
2003:
1915:
1747:
1137:
1081:
1052:
1032:
1012:
852:
416:
392:
365:
345:
325:
305:
285:
265:
245:
225:
205:
185:
165:
145:
125:
105:
1807:
39:
is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given
3771:
2015:
3754:
2033:
Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of
3804:
3284:
3120:
27:
of a semiorder. Two items are comparable when their vertical coordinates differ by at least one unit (the spacing between solid blue lines).
3601:
3076:, Theory and Decision Library. Series B: Mathematical and Statistical Methods, vol. 36, Dordrecht: Kluwer Academic Publishers Group,
1923:
3737:
3596:
3081:
1186:
then there do not exist sufficiently many sufficiently well-spaced real-numbers for it to be representable by a utility function.
3591:
1597:
is a partial order that has been constructed in this way from a semiorder. Then the semiorder may be recovered by declaring that
2291:
and the definition of incomparability, while each of the remaining two is equivalent to one of the above prohibition properties.
3227:
3309:
3035:
1809:, so every semiorder is an example of an interval order. A relation is a semiorder if, and only if, it can be obtained as an
83:
The original motivation for introducing semiorders was to model human preferences without assuming that incomparability is a
3628:
3548:
3222:
1562:
defined in this way meets the three requirements of a partial order that it be reflexive, antisymmetric, and transitive.
3413:
3342:
2859:
550:
forms a semiorder. If, instead, objects are declared comparable whenever their utilities differ, the result would be a
3316:
3304:
3267:
3242:
3217:
3171:
3140:
1721:
weak ordering, since the rightmost vertex is incomparable to its two closest left neighbors, but they are comparable.
3247:
3237:
3613:
3113:
3586:
3252:
3518:
3145:
1886:
2805:
Jamison, Dean T.; Lau, Lawrence J. (October 1977), "The nature of equilibrium with semiordered preferences",
1816:
3766:
3749:
48:
2041:
63:, in which items with equal scores may be tied but there is no margin of error. They are a special case of
3678:
3294:
3809:
3656:
3491:
3482:
3351:
3186:
3150:
3106:
2766:
Jamison, Dean T.; Lau, Lawrence J. (Sep–Nov 1975), "Semiorders and the Theory of Choice: A Correction",
596:
3232:
2549:
Chandon, J.-L.; Lemaire, J.; Pouget, J. (1978), "DĂ©nombrement des quasi-ordres sur un ensemble fini",
3744:
3703:
3693:
3683:
3428:
3391:
3381:
3361:
3346:
2598:
471:
60:
2967:
Proof
Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968)
2551:
Centre de Mathématique
Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
3671:
3582:
3528:
3487:
3477:
3366:
3299:
3262:
3027:
2288:
2195:
927:
857:
728:
658:
84:
3710:
3563:
3472:
3462:
3403:
3321:
3052:
3006:
2882:
2822:
2783:
2754:
2537:
2199:
805:
207:
is halfway between the two of them. Then, a person who desires more of the material would prefer
3783:
3623:
3257:
1568:
1533:
1209:
3720:
3698:
3558:
3543:
3523:
3326:
3077:
2696:
2664:
2632:
2515:
1678:
1241:
1150:
1086:
551:
521:
1652:
1626:
1507:
1481:
1455:
1403:
1377:
1351:
1273:
3533:
3386:
3044:
2998:
2940:
2911:
2874:
2814:
2775:
2746:
2708:
2676:
2644:
2610:
2575:
2529:
2495:
2034:
1600:
1299:
971:
901:
772:
702:
632:
606:
445:
395:
371:
3091:
3018:
2974:
2954:
2923:
2894:
2847:
2722:
2688:
2656:
2624:
2589:
2558:
2507:
554:, for which incomparability of objects (based on equality of numbers) would be transitive.
3715:
3498:
3376:
3371:
3356:
3181:
3166:
3087:
3014:
2970:
2962:
2950:
2919:
2890:
2843:
2734:
2718:
2684:
2652:
2620:
2585:
2554:
2520:
2503:
2027:
1882:
1878:
1054:. So it is impossible to have a three-point linear order with a fourth incomparable point.
425:
40:
3272:
2151:: if two semiorders on the same set differ from each other by the addition or removal of
1429:
1325:
3633:
3618:
3608:
3467:
3445:
3423:
2986:
2855:
2203:
2174:
2154:
2130:
2107:
2087:
2067:
2047:
1988:
1918:
1900:
1874:
1810:
1750:
1732:
1190:
supplies a precise characterization of the semiorders that may be defined numerically.
1140:
1122:
1066:
1037:
1017:
997:
837:
401:
377:
350:
330:
310:
290:
270:
250:
230:
210:
190:
170:
150:
130:
110:
90:
68:
52:
2615:
1756:
3798:
3732:
3688:
3666:
3538:
3408:
3396:
3201:
2945:
2915:
2834:
Kim, K. H.; Roush, F. W. (1978), "Enumeration of isomorphism classes of semiorders",
2713:
2680:
2648:
2499:
2044:: in any finite semiorder that is not a total order, there exists a pair of elements
1204:
64:
44:
24:
2699:; Trotter, W. T. (1992), "Linear extensions of semiorders: a maximization problem",
2541:
3553:
3435:
3418:
3336:
3176:
3129:
2730:
796:
574:
32:
3759:
3452:
3331:
3196:
2799:, Stanford University, Institute for Mathematical Studies in the Social Sciences
1183:
1180:
419:
247:, but would not have a preference between the other two pairs. In this example,
2486:
Avery, Peter (1992), "An algorithmic proof that semiorders are representable",
3727:
3661:
3502:
2982:
1063:, can be taken as a combinatorial definition of semiorders. If a semiorder on
3778:
3651:
2580:
1530:) follows from the second semiorder axiom. Therefore, the binary relation
795:. Therefore, it is impossible to have two mutually incomparable two-point
3573:
3440:
3191:
2635:(1970), "Intransitive indifference with unequal indifference intervals",
603:
Whenever two disjoint pairs of elements are comparable, for instance as
588:
Forbidden: three linearly ordered points and a fourth incomparable point
3056:
3010:
2886:
2826:
2787:
2758:
2667:(1973), "Interval representations for interval orders and semiorders",
2533:
1348:. Of the axioms that a partial order is required to obey, reflexivity (
655:, there must be an additional comparison among these elements, because
582:
567:
370:
To model this mathematically, suppose that objects are given numerical
2797:
Semiorders, Revealed
Preference, and the Theory of the Consumer Demand
19:
2902:
Rabinovitch, Issie (1977), "The Scott-Suppes theorem on semiorders",
367:
are comparable, so incomparability does not obey the transitive law.
3048:
3002:
2878:
2818:
2779:
2750:
2124:
in between 1/3 and 2/3 of the linear extensions of the semiorder.
1060:
187:
by the smallest amount that is perceptible as a difference, while
72:
71:, and can be characterized among the partial orders by additional
18:
2801:. Presented at the World Economics Congress, Cambridge, Sep 1970.
2287:, p. 181) used four axioms, the first two of which combine
3102:
3098:
2363:", Sen's remark on p.314 is likely to mean the latter property.
2339:, Section 10, p. 314) Since Luce modelled indifference between
2008:
1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, ... (sequence
1374:) follows automatically from this definition. Antisymmetry (if
2566:
Dean, R. A.; Keller, Gordon (1968), "Natural partial orders",
595:
A semiorder, defined from a utility function as above, is a
2989:(1958), "Foundational aspects of theories of measurement",
2010:
1452:) follows from the first semiorder axiom. Transitivity (if
2931:
Rabinovitch, Issie (1978), "The dimension of semiorders",
2836:
147:
represent three quantities of the same material, and that
2251:
2249:
2247:
2245:
2171:
order relations, then it is possible to find a path of
3074:
Semiorders: Properties, representations, applications
2220:
2218:
2177:
2157:
2133:
2110:
2090:
2070:
2050:
1991:
1926:
1903:
1819:
1759:
1735:
1681:
1655:
1629:
1603:
1571:
1536:
1510:
1484:
1458:
1432:
1406:
1380:
1354:
1328:
1302:
1276:
1244:
1212:
1153:
1125:
1089:
1069:
1040:
1020:
1000:
974:
930:
904:
860:
840:
808:
775:
731:
705:
661:
635:
609:
524:
474:
448:
428:
404:
380:
353:
333:
313:
293:
273:
253:
233:
213:
193:
173:
153:
133:
113:
93:
2421:
854:
must be comparable to at least one of them, because
287:
are incomparable in the preference ordering, as are
3644:
3572:
3511:
3281:
3210:
3159:
2860:"Semiorders and a theory of utility discrimination"
2737:(Sep 1973), "Semiorders and the Theory of Choice",
2183:
2163:
2139:
2116:
2096:
2076:
2056:
1997:
1977:
1909:
1857:
1801:
1741:
1699:
1667:
1641:
1615:
1589:
1554:
1522:
1496:
1470:
1444:
1418:
1392:
1366:
1340:
1314:
1288:
1262:
1230:
1171:
1131:
1111:
1075:
1046:
1026:
1006:
986:
960:
916:
890:
846:
826:
787:
761:
717:
691:
647:
621:
542:
510:
460:
434:
410:
386:
359:
339:
319:
299:
279:
259:
239:
219:
199:
179:
159:
139:
119:
99:
59:) as a model of human preference. They generalize
2518:(1989), "Semiorders and the 1/3–2/3 conjecture",
1978:{\displaystyle {\frac {1}{n+1}}{\binom {2n}{n}},}
1966:
1948:
2795:Jamison, Dean T.; Lau, Lawrence J. (July 1970),
2457:
2445:
573:Forbidden: two mutually incomparable two-point
2969:, Academic Press, New York, pp. 139–146,
1729:The semiorder defined from a utility function
3114:
2601:(1997), "Well-graded families of relations",
1179:(as defined by forbidden orders) includes an
8:
398:that maps the objects to be compared (a set
47:. Semiorders were introduced and applied in
2433:
2272:
2268:
75:, or by two forbidden four-item suborders.
3772:Positive cone of a partially ordered group
3121:
3107:
3099:
3028:"Choice Functions and Revealed Preference"
2406:
2372:
2255:
2944:
2933:Journal of Combinatorial Theory, Series A
2712:
2614:
2579:
2176:
2156:
2132:
2109:
2089:
2069:
2049:
1990:
1965:
1947:
1945:
1927:
1925:
1902:
1840:
1827:
1818:
1758:
1734:
1680:
1654:
1628:
1602:
1570:
1535:
1509:
1483:
1457:
1431:
1405:
1379:
1353:
1327:
1301:
1275:
1243:
1211:
1152:
1124:
1100:
1088:
1068:
1039:
1019:
999:
973:
929:
903:
859:
839:
807:
802:If three elements form a linear ordering
774:
730:
704:
660:
634:
608:
523:
473:
447:
427:
403:
379:
352:
332:
312:
292:
272:
252:
232:
212:
192:
172:
152:
132:
112:
92:
16:Numerical ordering with a margin of error
3755:Positive cone of an ordered vector space
2410:
2324:
2312:
1187:
2469:
2214:
2005:labeled items is given by the sequence
1858:{\displaystyle (\ell _{i},\ell _{i}+1)}
2300:
2267:This result is typically credited to
1897:The number of distinct semiorders on
7:
2422:Chandon, Lemaire & Pouget (1978)
2284:
2236:
2224:
56:
2336:
1749:may equivalently be defined as the
599:with the following two properties:
3282:Properties & Types (
2904:Journal of Mathematical Psychology
2669:Journal of Mathematical Psychology
2637:Journal of Mathematical Psychology
2355:", while Sen modelled it as "both
1985:while the number of semiorders on
1952:
1885:and found to be a special case of
14:
3738:Positive cone of an ordered field
2040:Semiorders are known to obey the
3592:Ordered topological vector space
3072:Pirlot, M.; Vincke, Ph. (1997),
2202:, and are a special case of the
1917:unlabeled items is given by the
1194:Relation to other kinds of order
581:
566:
2965:(1969), "Indifference graphs",
2568:Canadian Journal of Mathematics
1877:, semi-orders were examined by
1709:
511:{\displaystyle u(x)\leq u(y)-1}
3805:Properties of binary relations
3036:The Review of Economic Studies
1852:
1820:
1796:
1787:
1781:
1772:
1766:
1760:
1694:
1682:
1584:
1572:
1549:
1537:
1257:
1245:
1225:
1213:
1166:
1154:
1106:
1093:
994:, in either case showing that
955:
949:
940:
934:
885:
879:
870:
864:
756:
750:
741:
735:
686:
680:
671:
665:
537:
525:
499:
493:
484:
478:
1:
3549:Series-parallel partial order
3026:Sen, Amartya K. (July 1971),
2991:The Journal of Symbolic Logic
2616:10.1016/S0012-365X(96)00095-7
2458:Doignon & Falmagne (1997)
2446:Fishburn & Trotter (1992)
961:{\displaystyle u(z)\geq u(x)}
891:{\displaystyle u(z)\leq u(x)}
762:{\displaystyle u(w)\geq u(y)}
692:{\displaystyle u(w)\leq u(y)}
87:. For instance, suppose that
35:, a branch of mathematics, a
3228:Cantor's isomorphism theorem
2946:10.1016/0097-3165(78)90030-4
2916:10.1016/0022-2496(77)90030-x
2714:10.1016/0012-365X(92)90036-F
2681:10.1016/0022-2496(73)90007-2
2649:10.1016/0022-2496(70)90062-3
2500:10.1016/0196-6774(92)90010-A
2127:The set of semiorders on an
3268:Szpilrajn extension theorem
3243:Hausdorff maximal principle
3218:Boolean prime ideal theorem
827:{\displaystyle w<x<y}
442:on the objects, by setting
3826:
3614:Topological vector lattice
834:, then every fourth point
3136:
2269:Scott & Suppes (1958)
2198:of semiorders are called
2026:Any finite semiorder has
1893:Combinatorial enumeration
1887:quasitransitive relations
1869:Quasitransitive relations
1813:of unit length intervals
1753:defined by the intervals
1590:{\displaystyle (X,\leq )}
1565:Conversely, suppose that
1555:{\displaystyle (X,\leq )}
1231:{\displaystyle (X,\leq )}
3223:Cantor–Bernstein theorem
2407:Dean & Keller (1968)
2393:more general, to adding
2373:Jamison & Lau (1970)
1700:{\displaystyle (X,<)}
1263:{\displaystyle (X,<)}
1172:{\displaystyle (X,<)}
1112:{\displaystyle O(n^{2})}
543:{\displaystyle (X,<)}
3767:Partially ordered group
3587:Specialization preorder
1668:{\displaystyle x\neq y}
1642:{\displaystyle x\leq y}
1523:{\displaystyle x\leq z}
1497:{\displaystyle y\leq z}
1471:{\displaystyle x\leq y}
1419:{\displaystyle y\leq x}
1393:{\displaystyle x\leq y}
1367:{\displaystyle x\leq x}
1289:{\displaystyle x\leq y}
49:mathematical psychology
3253:Kruskal's tree theorem
3248:Knaster–Tarski theorem
3238:Dushnik–Miller theorem
2581:10.4153/CJM-1968-055-7
2411:Kim & Roush (1978)
2384:since it is transitive
2196:incomparability graphs
2185:
2165:
2141:
2118:
2098:
2078:
2058:
1999:
1979:
1911:
1859:
1803:
1743:
1701:
1669:
1643:
1617:
1616:{\displaystyle x<y}
1591:
1556:
1524:
1498:
1472:
1446:
1420:
1394:
1368:
1342:
1316:
1315:{\displaystyle x<y}
1290:
1264:
1232:
1184:totally ordered subset
1173:
1133:
1113:
1077:
1048:
1028:
1008:
988:
987:{\displaystyle w<z}
962:
918:
917:{\displaystyle z<y}
892:
848:
828:
789:
788:{\displaystyle y<x}
763:
719:
718:{\displaystyle w<z}
693:
649:
648:{\displaystyle y<z}
623:
622:{\displaystyle w<x}
544:
512:
462:
461:{\displaystyle x<y}
436:
412:
388:
361:
341:
321:
301:
281:
261:
241:
221:
201:
181:
161:
141:
121:
101:
28:
2599:Falmagne, Jean-Claude
2516:Brightwell, Graham R.
2488:Journal of Algorithms
2186:
2166:
2142:
2119:
2104:appears earlier than
2099:
2079:
2059:
2000:
1980:
1912:
1860:
1804:
1744:
1702:
1670:
1644:
1618:
1592:
1557:
1525:
1499:
1473:
1447:
1421:
1395:
1369:
1343:
1317:
1291:
1265:
1233:
1174:
1134:
1114:
1078:
1049:
1029:
1009:
989:
963:
919:
893:
849:
829:
790:
764:
720:
694:
650:
624:
597:partially ordered set
545:
513:
463:
437:
413:
389:
362:
342:
322:
302:
282:
262:
242:
222:
202:
182:
162:
142:
122:
102:
61:strict weak orderings
22:
3745:Ordered vector space
2701:Discrete Mathematics
2603:Discrete Mathematics
2597:Doignon, Jean-Paul;
2175:
2155:
2131:
2108:
2088:
2068:
2048:
1989:
1924:
1901:
1817:
1757:
1733:
1679:
1653:
1627:
1601:
1569:
1534:
1508:
1482:
1456:
1430:
1404:
1378:
1352:
1326:
1300:
1274:
1242:
1210:
1151:
1123:
1087:
1067:
1038:
1018:
998:
972:
928:
902:
858:
838:
806:
773:
729:
703:
659:
633:
607:
552:strict weak ordering
522:
472:
446:
435:{\displaystyle <}
426:
402:
378:
351:
331:
311:
291:
271:
251:
231:
211:
191:
171:
151:
131:
111:
91:
3583:Alexandrov topology
3529:Lexicographic order
3488:Well-quasi-ordering
2200:indifference graphs
1445:{\displaystyle x=y}
1341:{\displaystyle x=y}
85:transitive relation
3564:Transitive closure
3524:Converse/Transpose
3233:Dilworth's theorem
2697:Fishburn, Peter C.
2665:Fishburn, Peter C.
2633:Fishburn, Peter C.
2534:10.1007/BF00353656
2434:Rabinovitch (1978)
2397:symmetric relation
2273:Rabinovitch (1977)
2181:
2161:
2137:
2114:
2094:
2074:
2054:
2042:1/3–2/3 conjecture
1995:
1975:
1907:
1855:
1799:
1739:
1697:
1665:
1639:
1613:
1587:
1552:
1520:
1494:
1468:
1442:
1416:
1390:
1364:
1338:
1312:
1286:
1270:by declaring that
1260:
1228:
1169:
1139:is an instance of
1129:
1109:
1073:
1044:
1024:
1004:
984:
958:
914:
888:
844:
824:
785:
759:
715:
689:
645:
619:
540:
508:
458:
432:
408:
384:
357:
337:
317:
297:
277:
257:
237:
217:
197:
177:
157:
137:
117:
97:
29:
3792:
3791:
3750:Partially ordered
3559:Symmetric closure
3544:Reflexive closure
3287:
2553:(62): 61–80, 83,
2256:Brightwell (1989)
2184:{\displaystyle k}
2164:{\displaystyle k}
2140:{\displaystyle n}
2117:{\displaystyle y}
2097:{\displaystyle x}
2077:{\displaystyle y}
2057:{\displaystyle x}
2035:linear extensions
1998:{\displaystyle n}
1964:
1943:
1910:{\displaystyle n}
1742:{\displaystyle u}
1238:from a semiorder
1203:One may define a
1132:{\displaystyle O}
1076:{\displaystyle n}
1047:{\displaystyle y}
1027:{\displaystyle w}
1014:is comparable to
1007:{\displaystyle z}
847:{\displaystyle z}
411:{\displaystyle X}
387:{\displaystyle u}
360:{\displaystyle z}
340:{\displaystyle x}
320:{\displaystyle z}
300:{\displaystyle y}
280:{\displaystyle y}
260:{\displaystyle x}
240:{\displaystyle z}
220:{\displaystyle x}
200:{\displaystyle y}
180:{\displaystyle z}
160:{\displaystyle x}
140:{\displaystyle z}
120:{\displaystyle y}
100:{\displaystyle x}
3817:
3534:Linear extension
3283:
3263:Mirsky's theorem
3123:
3116:
3109:
3100:
3094:
3059:
3032:
3021:
2977:
2963:Roberts, Fred S.
2957:
2948:
2926:
2897:
2864:
2850:
2829:
2813:(7): 1595–1605,
2800:
2790:
2774:(5–6): 979–980,
2761:
2735:Lau, Lawrence J.
2731:Jamison, Dean T.
2725:
2716:
2691:
2659:
2627:
2618:
2592:
2583:
2561:
2544:
2510:
2473:
2467:
2461:
2455:
2449:
2443:
2437:
2431:
2425:
2419:
2413:
2404:
2398:
2391:
2385:
2382:
2376:
2370:
2364:
2334:
2328:
2322:
2316:
2310:
2304:
2298:
2292:
2282:
2276:
2265:
2259:
2253:
2240:
2234:
2228:
2222:
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2147:-element set is
2146:
2144:
2143:
2138:
2123:
2121:
2120:
2115:
2103:
2101:
2100:
2095:
2083:
2081:
2080:
2075:
2063:
2061:
2060:
2055:
2037:are semiorders.
2013:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1971:
1970:
1969:
1960:
1951:
1944:
1942:
1928:
1916:
1914:
1913:
1908:
1864:
1862:
1861:
1856:
1845:
1844:
1832:
1831:
1808:
1806:
1805:
1802:{\displaystyle }
1800:
1748:
1746:
1745:
1740:
1706:
1704:
1703:
1698:
1674:
1672:
1671:
1666:
1648:
1646:
1645:
1640:
1622:
1620:
1619:
1614:
1596:
1594:
1593:
1588:
1561:
1559:
1558:
1553:
1529:
1527:
1526:
1521:
1503:
1501:
1500:
1495:
1477:
1475:
1474:
1469:
1451:
1449:
1448:
1443:
1425:
1423:
1422:
1417:
1399:
1397:
1396:
1391:
1373:
1371:
1370:
1365:
1347:
1345:
1344:
1339:
1321:
1319:
1318:
1313:
1296:whenever either
1295:
1293:
1292:
1287:
1269:
1267:
1266:
1261:
1237:
1235:
1234:
1229:
1178:
1176:
1175:
1170:
1138:
1136:
1135:
1130:
1118:
1116:
1115:
1110:
1105:
1104:
1082:
1080:
1079:
1074:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
967:
965:
964:
959:
923:
921:
920:
915:
897:
895:
894:
889:
853:
851:
850:
845:
833:
831:
830:
825:
794:
792:
791:
786:
768:
766:
765:
760:
724:
722:
721:
716:
698:
696:
695:
690:
654:
652:
651:
646:
628:
626:
625:
620:
585:
570:
549:
547:
546:
541:
517:
515:
514:
509:
467:
465:
464:
459:
441:
439:
438:
433:
417:
415:
414:
409:
396:utility function
393:
391:
390:
385:
366:
364:
363:
358:
346:
344:
343:
338:
326:
324:
323:
318:
306:
304:
303:
298:
286:
284:
283:
278:
266:
264:
263:
258:
246:
244:
243:
238:
226:
224:
223:
218:
206:
204:
203:
198:
186:
184:
183:
178:
166:
164:
163:
158:
146:
144:
143:
138:
126:
124:
123:
118:
106:
104:
103:
98:
3825:
3824:
3820:
3819:
3818:
3816:
3815:
3814:
3795:
3794:
3793:
3788:
3784:Young's lattice
3640:
3568:
3507:
3357:Heyting algebra
3305:Boolean algebra
3277:
3258:Laver's theorem
3206:
3172:Boolean algebra
3167:Binary relation
3155:
3132:
3127:
3084:
3071:
3068:
3066:Further reading
3063:
3049:10.2307/2296384
3030:
3025:
3003:10.2307/2964389
2987:Suppes, Patrick
2981:
2961:
2930:
2901:
2879:10.2307/1905751
2862:
2856:Luce, R. Duncan
2854:
2833:
2819:10.2307/1913952
2804:
2794:
2780:10.2307/1911339
2765:
2751:10.2307/1913813
2729:
2695:
2663:
2631:
2596:
2565:
2548:
2514:
2485:
2481:
2476:
2468:
2464:
2456:
2452:
2444:
2440:
2432:
2428:
2420:
2416:
2405:
2401:
2392:
2388:
2383:
2379:
2371:
2367:
2335:
2331:
2325:Fishburn (1970)
2323:
2319:
2313:Fishburn (1973)
2311:
2307:
2299:
2295:
2283:
2279:
2266:
2262:
2254:
2243:
2235:
2231:
2223:
2216:
2212:
2204:interval graphs
2173:
2172:
2153:
2152:
2129:
2128:
2106:
2105:
2086:
2085:
2066:
2065:
2046:
2045:
2030:at most three.
2028:order dimension
2024:
2019:
2009:
1987:
1986:
1953:
1946:
1932:
1922:
1921:
1919:Catalan numbers
1899:
1898:
1895:
1883:Lawrence J. Lau
1879:Dean T. Jamison
1871:
1836:
1823:
1815:
1814:
1755:
1754:
1731:
1730:
1727:
1725:Interval orders
1718:
1677:
1676:
1651:
1650:
1625:
1624:
1599:
1598:
1567:
1566:
1532:
1531:
1506:
1505:
1480:
1479:
1454:
1453:
1428:
1427:
1402:
1401:
1376:
1375:
1350:
1349:
1324:
1323:
1298:
1297:
1272:
1271:
1240:
1239:
1208:
1207:
1201:
1196:
1188:Fishburn (1973)
1149:
1148:
1121:
1120:
1096:
1085:
1084:
1065:
1064:
1036:
1035:
1016:
1015:
996:
995:
970:
969:
926:
925:
900:
899:
856:
855:
836:
835:
804:
803:
771:
770:
727:
726:
701:
700:
657:
656:
631:
630:
605:
604:
593:
592:
591:
590:
589:
586:
578:
577:
571:
560:
520:
519:
470:
469:
444:
443:
424:
423:
400:
399:
376:
375:
349:
348:
329:
328:
309:
308:
289:
288:
269:
268:
249:
248:
229:
228:
209:
208:
189:
188:
169:
168:
167:is larger than
149:
148:
129:
128:
109:
108:
89:
88:
81:
69:interval orders
53:Duncan Luce
41:margin of error
17:
12:
11:
5:
3823:
3821:
3813:
3812:
3807:
3797:
3796:
3790:
3789:
3787:
3786:
3781:
3776:
3775:
3774:
3764:
3763:
3762:
3757:
3752:
3742:
3741:
3740:
3730:
3725:
3724:
3723:
3718:
3711:Order morphism
3708:
3707:
3706:
3696:
3691:
3686:
3681:
3676:
3675:
3674:
3664:
3659:
3654:
3648:
3646:
3642:
3641:
3639:
3638:
3637:
3636:
3631:
3629:Locally convex
3626:
3621:
3611:
3609:Order topology
3606:
3605:
3604:
3602:Order topology
3599:
3589:
3579:
3577:
3570:
3569:
3567:
3566:
3561:
3556:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3515:
3513:
3509:
3508:
3506:
3505:
3495:
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3449:
3448:
3438:
3433:
3432:
3431:
3426:
3421:
3416:
3414:Chain-complete
3406:
3401:
3400:
3399:
3394:
3389:
3384:
3379:
3369:
3364:
3359:
3354:
3349:
3339:
3334:
3329:
3324:
3319:
3314:
3313:
3312:
3302:
3297:
3291:
3289:
3279:
3278:
3276:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3225:
3220:
3214:
3212:
3208:
3207:
3205:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3163:
3161:
3157:
3156:
3154:
3153:
3148:
3143:
3137:
3134:
3133:
3128:
3126:
3125:
3118:
3111:
3103:
3097:
3096:
3082:
3067:
3064:
3062:
3061:
3043:(3): 307–317,
3023:
2997:(2): 113–128,
2979:
2959:
2928:
2910:(2): 209–212,
2899:
2873:(2): 178–191,
2852:
2831:
2802:
2792:
2763:
2745:(5): 901–912,
2727:
2693:
2661:
2629:
2609:(1–3): 35–44,
2594:
2563:
2546:
2528:(4): 369–380,
2512:
2494:(1): 144–147,
2482:
2480:
2477:
2475:
2474:
2470:Roberts (1969)
2462:
2450:
2438:
2426:
2414:
2399:
2386:
2377:
2365:
2329:
2317:
2305:
2293:
2277:
2260:
2241:
2229:
2227:, p. 179.
2213:
2211:
2208:
2180:
2160:
2136:
2113:
2093:
2073:
2053:
2023:
2020:
2007:
1994:
1974:
1968:
1963:
1959:
1956:
1950:
1941:
1938:
1935:
1931:
1906:
1894:
1891:
1875:Amartya K. Sen
1870:
1867:
1854:
1851:
1848:
1843:
1839:
1835:
1830:
1826:
1822:
1811:interval order
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1751:interval order
1738:
1726:
1723:
1717:
1714:
1696:
1693:
1690:
1687:
1684:
1664:
1661:
1658:
1638:
1635:
1632:
1612:
1609:
1606:
1586:
1583:
1580:
1577:
1574:
1551:
1548:
1545:
1542:
1539:
1519:
1516:
1513:
1493:
1490:
1487:
1467:
1464:
1461:
1441:
1438:
1435:
1415:
1412:
1409:
1389:
1386:
1383:
1363:
1360:
1357:
1337:
1334:
1331:
1311:
1308:
1305:
1285:
1282:
1279:
1259:
1256:
1253:
1250:
1247:
1227:
1224:
1221:
1218:
1215:
1200:
1199:Partial orders
1197:
1195:
1192:
1168:
1165:
1162:
1159:
1156:
1141:big O notation
1128:
1108:
1103:
1099:
1095:
1092:
1072:
1056:
1055:
1043:
1023:
1003:
983:
980:
977:
957:
954:
951:
948:
945:
942:
939:
936:
933:
913:
910:
907:
887:
884:
881:
878:
875:
872:
869:
866:
863:
843:
823:
820:
817:
814:
811:
800:
784:
781:
778:
758:
755:
752:
749:
746:
743:
740:
737:
734:
714:
711:
708:
688:
685:
682:
679:
676:
673:
670:
667:
664:
644:
641:
638:
618:
615:
612:
587:
580:
579:
572:
565:
564:
563:
562:
561:
559:
556:
539:
536:
533:
530:
527:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
457:
454:
451:
431:
407:
383:
372:utility values
356:
336:
316:
296:
276:
256:
236:
216:
196:
176:
156:
136:
116:
96:
80:
79:Utility theory
77:
65:partial orders
15:
13:
10:
9:
6:
4:
3:
2:
3822:
3811:
3808:
3806:
3803:
3802:
3800:
3785:
3782:
3780:
3777:
3773:
3770:
3769:
3768:
3765:
3761:
3758:
3756:
3753:
3751:
3748:
3747:
3746:
3743:
3739:
3736:
3735:
3734:
3733:Ordered field
3731:
3729:
3726:
3722:
3719:
3717:
3714:
3713:
3712:
3709:
3705:
3702:
3701:
3700:
3697:
3695:
3692:
3690:
3689:Hasse diagram
3687:
3685:
3682:
3680:
3677:
3673:
3670:
3669:
3668:
3667:Comparability
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3649:
3647:
3643:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3615:
3612:
3610:
3607:
3603:
3600:
3598:
3595:
3594:
3593:
3590:
3588:
3584:
3581:
3580:
3578:
3575:
3571:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3539:Product order
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3516:
3514:
3512:Constructions
3510:
3504:
3500:
3496:
3493:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3447:
3444:
3443:
3442:
3439:
3437:
3434:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3411:
3410:
3409:Partial order
3407:
3405:
3402:
3398:
3397:Join and meet
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3374:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3344:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3311:
3308:
3307:
3306:
3303:
3301:
3298:
3296:
3295:Antisymmetric
3293:
3292:
3290:
3286:
3280:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3215:
3213:
3209:
3203:
3202:Weak ordering
3200:
3198:
3195:
3193:
3190:
3188:
3187:Partial order
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3164:
3162:
3158:
3152:
3149:
3147:
3144:
3142:
3139:
3138:
3135:
3131:
3124:
3119:
3117:
3112:
3110:
3105:
3104:
3101:
3093:
3089:
3085:
3083:0-7923-4617-3
3079:
3075:
3070:
3069:
3065:
3058:
3054:
3050:
3046:
3042:
3038:
3037:
3029:
3024:
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2947:
2942:
2938:
2934:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2900:
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2803:
2798:
2793:
2789:
2785:
2781:
2777:
2773:
2769:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2715:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2617:
2612:
2608:
2604:
2600:
2595:
2591:
2587:
2582:
2577:
2573:
2569:
2564:
2560:
2556:
2552:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2522:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2489:
2484:
2483:
2478:
2471:
2466:
2463:
2459:
2454:
2451:
2447:
2442:
2439:
2435:
2430:
2427:
2423:
2418:
2415:
2412:
2408:
2403:
2400:
2396:
2390:
2387:
2381:
2378:
2374:
2369:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2333:
2330:
2326:
2321:
2318:
2314:
2309:
2306:
2302:
2297:
2294:
2290:
2286:
2281:
2278:
2274:
2271:; see, e.g.,
2270:
2264:
2261:
2257:
2252:
2250:
2248:
2246:
2242:
2238:
2233:
2230:
2226:
2221:
2219:
2215:
2209:
2207:
2205:
2201:
2197:
2192:
2178:
2158:
2150:
2134:
2125:
2111:
2091:
2071:
2051:
2043:
2038:
2036:
2031:
2029:
2022:Other results
2021:
2017:
2012:
2006:
1992:
1972:
1961:
1957:
1954:
1939:
1936:
1933:
1929:
1920:
1904:
1892:
1890:
1888:
1884:
1880:
1876:
1873:According to
1868:
1866:
1849:
1846:
1841:
1837:
1833:
1828:
1824:
1812:
1793:
1790:
1784:
1778:
1775:
1769:
1763:
1752:
1736:
1724:
1722:
1715:
1713:
1711:
1691:
1688:
1685:
1662:
1659:
1656:
1636:
1633:
1630:
1610:
1607:
1604:
1581:
1578:
1575:
1563:
1546:
1543:
1540:
1517:
1514:
1511:
1491:
1488:
1485:
1465:
1462:
1459:
1439:
1436:
1433:
1413:
1410:
1407:
1387:
1384:
1381:
1361:
1358:
1355:
1335:
1332:
1329:
1309:
1306:
1303:
1283:
1280:
1277:
1254:
1251:
1248:
1222:
1219:
1216:
1206:
1205:partial order
1198:
1193:
1191:
1189:
1185:
1182:
1163:
1160:
1157:
1144:
1142:
1126:
1101:
1097:
1090:
1070:
1062:
1041:
1021:
1001:
981:
978:
975:
952:
946:
943:
937:
931:
911:
908:
905:
882:
876:
873:
867:
861:
841:
821:
818:
815:
812:
809:
801:
798:
797:linear orders
782:
779:
776:
753:
747:
744:
738:
732:
712:
709:
706:
683:
677:
674:
668:
662:
642:
639:
636:
616:
613:
610:
602:
601:
600:
598:
584:
576:
575:linear orders
569:
557:
555:
553:
534:
531:
528:
505:
502:
496:
490:
487:
481:
475:
455:
452:
449:
429:
421:
405:
397:
381:
374:, by letting
373:
368:
354:
334:
314:
294:
274:
254:
234:
214:
194:
174:
154:
134:
114:
94:
86:
78:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
26:
25:Hasse diagram
21:
3810:Order theory
3576:& Orders
3554:Star product
3483:Well-founded
3457:
3436:Prefix order
3392:Distributive
3382:Complemented
3352:Foundational
3317:Completeness
3273:Zorn's lemma
3177:Cyclic order
3160:Key concepts
3130:Order theory
3073:
3040:
3034:
2994:
2990:
2966:
2939:(1): 50–61,
2936:
2932:
2907:
2903:
2870:
2867:Econometrica
2866:
2842:(2): 58–61,
2839:
2835:
2810:
2807:Econometrica
2806:
2796:
2771:
2768:Econometrica
2767:
2742:
2739:Econometrica
2738:
2707:(1): 25–40,
2704:
2700:
2672:
2668:
2640:
2636:
2606:
2602:
2571:
2567:
2550:
2525:
2519:
2491:
2487:
2465:
2453:
2441:
2429:
2417:
2402:
2394:
2389:
2380:
2368:
2360:
2356:
2352:
2348:
2347:as "neither
2344:
2340:
2332:
2320:
2308:
2301:Avery (1992)
2296:
2280:
2263:
2232:
2193:
2148:
2126:
2039:
2032:
2025:
1896:
1872:
1728:
1719:
1564:
1202:
1145:
1119:, where the
1057:
968:would imply
898:would imply
769:would imply
699:would imply
594:
420:real numbers
369:
82:
45:incomparable
36:
33:order theory
30:
3760:Riesz space
3721:Isomorphism
3597:Normal cone
3519:Composition
3453:Semilattice
3362:Homogeneous
3347:Equivalence
3197:Total order
2983:Scott, Dana
2643:: 144–149,
2574:: 535–554,
2237:Luce (1956)
2225:Luce (1956)
2149:well-graded
1716:Weak orders
1710:#Axiomatics
1181:uncountable
43:are deemed
3799:Categories
3728:Order type
3662:Cofinality
3503:Well-order
3478:Transitive
3367:Idempotent
3300:Asymmetric
2675:: 91–105,
2479:References
2285:Luce (1956
2084:such that
558:Axiomatics
3779:Upper set
3716:Embedding
3652:Antichain
3473:Tolerance
3463:Symmetric
3458:Semiorder
3404:Reflexive
3322:Connected
2337:Sen (1971
2289:asymmetry
1838:ℓ
1825:ℓ
1660:≠
1634:≤
1623:whenever
1582:≤
1547:≤
1515:≤
1489:≤
1463:≤
1411:≤
1385:≤
1359:≤
1281:≤
1223:≤
944:≥
874:≤
745:≥
675:≤
503:−
488:≤
468:whenever
37:semiorder
3574:Topology
3441:Preorder
3424:Eulerian
3387:Complete
3337:Directed
3327:Covering
3192:Preorder
3151:Category
3146:Glossary
2858:(1956),
2542:86860160
3679:Duality
3657:Cofinal
3645:Related
3624:Fréchet
3501:)
3377:Bounded
3372:Lattice
3345:)
3343:Partial
3211:Results
3182:Lattice
3092:1472236
3057:2296384
3019:0115919
3011:2964389
2975:0252267
2955:0498294
2924:0437404
2895:0078632
2887:1905751
2848:0538212
2827:1913952
2788:1911339
2759:1913813
2723:1171114
2689:0316322
2657:0253942
2625:1468838
2590:0225686
2559:0517680
2508:1146337
2014:in the
2011:A006531
518:. Then
394:be any
67:and of
55: (
3704:Subnet
3684:Filter
3634:Normed
3619:Banach
3585:&
3492:Better
3429:Strict
3419:Graded
3310:topics
3141:Topics
3090:
3080:
3055:
3017:
3009:
2973:
2953:
2922:
2893:
2885:
2846:
2825:
2786:
2757:
2721:
2687:
2655:
2623:
2588:
2557:
2540:
2506:
1061:axioms
1034:or to
924:while
725:while
327:, but
127:, and
73:axioms
3694:Ideal
3672:Graph
3468:Total
3446:Total
3332:Dense
3053:JSTOR
3031:(PDF)
3007:JSTOR
2883:JSTOR
2863:(PDF)
2823:JSTOR
2784:JSTOR
2755:JSTOR
2538:S2CID
2521:Order
2210:Notes
1504:then
1426:then
418:) to
3285:list
3078:ISBN
2359:and
2351:nor
2343:and
2194:The
2064:and
2016:OEIS
1881:and
1692:<
1649:and
1608:<
1478:and
1400:and
1307:<
1255:<
1164:<
979:<
909:<
819:<
813:<
780:<
710:<
640:<
629:and
614:<
535:<
453:<
430:<
347:and
307:and
267:and
57:1956
23:The
3699:Net
3499:Pre
3045:doi
2999:doi
2941:doi
2912:doi
2875:doi
2815:doi
2776:doi
2747:doi
2709:doi
2705:103
2677:doi
2645:doi
2611:doi
2607:173
2576:doi
2530:doi
2496:doi
2395:any
2361:yRx
2357:xRy
2353:yRx
2349:xRy
1712:).
1322:or
227:to
51:by
31:In
3801::
3088:MR
3086:,
3051:,
3041:38
3039:,
3033:,
3015:MR
3013:,
3005:,
2995:23
2993:,
2985:;
2971:MR
2951:MR
2949:,
2937:25
2935:,
2920:MR
2918:,
2908:15
2906:,
2891:MR
2889:,
2881:,
2871:24
2869:,
2865:,
2844:MR
2838:,
2821:,
2811:45
2809:,
2782:,
2772:43
2770:,
2753:,
2743:41
2741:,
2733:;
2719:MR
2717:,
2703:,
2685:MR
2683:,
2673:10
2671:,
2653:MR
2651:,
2639:,
2621:MR
2619:,
2605:,
2586:MR
2584:,
2572:20
2570:,
2555:MR
2536:,
2524:,
2504:MR
2502:,
2492:13
2490:,
2409:;
2244:^
2217:^
2206:.
1865:.
1143:.
107:,
3497:(
3494:)
3490:(
3341:(
3288:)
3122:e
3115:t
3108:v
3095:.
3060:.
3047::
3022:.
3001::
2978:.
2958:.
2943::
2927:.
2914::
2898:.
2877::
2851:.
2840:3
2830:.
2817::
2791:.
2778::
2762:.
2749::
2726:.
2711::
2692:.
2679::
2660:.
2647::
2641:7
2628:.
2613::
2593:.
2578::
2562:.
2545:.
2532::
2526:5
2511:.
2498::
2472:.
2460:.
2448:.
2436:.
2424:.
2375:.
2345:y
2341:x
2327:.
2315:.
2303:.
2275:.
2258:.
2179:k
2159:k
2135:n
2112:y
2092:x
2072:y
2052:x
2018:)
1993:n
1973:,
1967:)
1962:n
1958:n
1955:2
1949:(
1940:1
1937:+
1934:n
1930:1
1905:n
1853:)
1850:1
1847:+
1842:i
1834:,
1829:i
1821:(
1797:]
1794:1
1791:+
1788:)
1785:x
1782:(
1779:u
1776:,
1773:)
1770:x
1767:(
1764:u
1761:[
1737:u
1695:)
1689:,
1686:X
1683:(
1663:y
1657:x
1637:y
1631:x
1611:y
1605:x
1585:)
1579:,
1576:X
1573:(
1550:)
1544:,
1541:X
1538:(
1518:z
1512:x
1492:z
1486:y
1466:y
1460:x
1440:y
1437:=
1434:x
1414:x
1408:y
1388:y
1382:x
1362:x
1356:x
1336:y
1333:=
1330:x
1310:y
1304:x
1284:y
1278:x
1258:)
1252:,
1249:X
1246:(
1226:)
1220:,
1217:X
1214:(
1167:)
1161:,
1158:X
1155:(
1127:O
1107:)
1102:2
1098:n
1094:(
1091:O
1071:n
1042:y
1022:w
1002:z
982:z
976:w
956:)
953:x
950:(
947:u
941:)
938:z
935:(
932:u
912:y
906:z
886:)
883:x
880:(
877:u
871:)
868:z
865:(
862:u
842:z
822:y
816:x
810:w
799:.
783:x
777:y
757:)
754:y
751:(
748:u
742:)
739:w
736:(
733:u
713:z
707:w
687:)
684:y
681:(
678:u
672:)
669:w
666:(
663:u
643:z
637:y
617:x
611:w
538:)
532:,
529:X
526:(
506:1
500:)
497:y
494:(
491:u
485:)
482:x
479:(
476:u
456:y
450:x
406:X
382:u
355:z
335:x
315:z
295:y
275:y
255:x
235:z
215:x
195:y
175:z
155:x
135:z
115:y
95:x
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