1032:
3445:
69:
61:
53:
1040:
3481:
726:
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not
2858:
as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not
3626:. In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an
1023:). In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.
3389:
2050:
3138:
Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are
156:
if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.
2517:
706:
393:
285:
656:
606:
1081:
3803:
2780:
3100:
3004:
1400:
1526:
2433:
1573:
836:
A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if
788:
1015:) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a
3216:
2584:
1157:
3667:
The monotonic
Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators
1613:
2333:
2253:
1200:
3167:
2294:
1331:
1177:
343:
235:
936:
869:
556:
530:
2928:
1924:
1462:
1435:
420:
3269:
2905:
2885:
2832:
440:
2360:
2799:
2700:
2551:
2394:
2204:
2184:
2160:
2136:
2112:
2088:
1976:
1944:
1888:
1865:
1831:
1811:
1791:
1771:
1751:
1724:
1704:
1681:
1633:
1485:
1248:
1222:
1134:
1104:
808:
748:
504:
484:
305:
209:
189:
142:
2680:
2634:
986:
901:
4882:
3777:
1298:
3304:
4865:
2209:
The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the
833:
However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).
4395:
97:
999:, so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.
4231:
1988:
2444:
4145:
4126:
4107:
1255:
4712:
2707:
4848:
4707:
4088:
4069:
4050:
3949:
3911:
3044:
4014:
2951:
1340:
4702:
3429:
1983:
1403:
1258:
in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (
4338:
4420:
4184:
661:
348:
240:
611:
561:
4739:
4659:
4173:
1050:
4333:
4915:
4524:
4453:
3767:
4930:
4427:
4415:
4378:
4353:
4328:
4282:
4251:
4168:
4358:
4348:
4724:
4224:
3793:
39:
4697:
4363:
2519:
46:
4629:
4256:
3772:
2374:
4877:
4860:
3505:
1494:
4925:
4789:
4405:
3020:
2805:
if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator
2399:
1684:
1576:
93:
35:
2206:. In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.
4920:
4767:
4602:
4593:
4462:
4297:
4261:
4217:
3798:
3788:
3758:
task, can be achieved efficiently when all involved functions and predicates are monotonic and
Boolean.
3650:
2851:
1656:
4343:
4163:
2907:
are of little use in many non-total orders and hence no additional terminology is introduced for them.
757:
4855:
4814:
4804:
4794:
4539:
4502:
4492:
4472:
4457:
4180:
3587:
3417:
2264:
1111:
1020:
3174:
1531:
4782:
4693:
4639:
4598:
4588:
4477:
4410:
4373:
3675:
3669:
3425:
3421:
3394:
2556:
2370:
2260:
2139:
1107:
463:
4674:
4583:
4573:
4514:
4432:
3994:
3875:
3783:
3239:
2115:
1955:
1225:
1139:
4894:
4734:
4368:
1585:
2299:
4831:
4809:
4669:
4654:
4634:
4437:
4192:
4141:
4122:
4103:
4084:
4065:
4046:
3945:
3907:
3219:
3116:
2336:
2226:
2063:
2057:
1730:
1267:
1182:
4083:. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356.
3146:
2270:
1303:
1162:
322:
214:
4644:
4497:
4004:
3235:
2163:
1947:
1834:
1407:
1016:
906:
839:
827:
535:
509:
2913:
1893:
1440:
1413:
1031:
830:
because they are guaranteed to have a one-to-one mapping from their range to their domain.
405:
4826:
4609:
4487:
4482:
4467:
4292:
4277:
3741:
3245:
3140:
3128:
2890:
2870:
2808:
2523:
1979:
1841:
1334:
425:
4383:
3988:
2342:
1655:
These properties are the reason why monotonic functions are useful in technical work in
4744:
4729:
4719:
4578:
4556:
4534:
4195:
3169:
2784:
2685:
2536:
2379:
2189:
2169:
2145:
2121:
2097:
2073:
1961:
1929:
1873:
1850:
1816:
1796:
1776:
1756:
1736:
1709:
1689:
1666:
1618:
1488:
1470:
1233:
1207:
1119:
1089:
793:
733:
489:
469:
290:
194:
174:
127:
101:
31:
3842:
3444:
2639:
2593:
941:
874:
68:
60:
52:
4909:
4843:
4799:
4777:
4649:
4519:
4507:
4312:
3646:
3993:. Proc. 29th AAAI Conf. on Artificial Intelligence. AAAI Press. pp. 3702–3709.
1035:
Monotonic function with a dense set of jump discontinuities (several sections shown)
4664:
4546:
4529:
4447:
4287:
4240:
3654:
2864:
2527:
1837:
149:
109:
17:
450:). Again, by inverting the order symbol, one finds a corresponding concept called
30:"Monotonicity" redirects here. For information on monotonicity as it pertains to
4870:
4563:
4442:
4307:
3751:
2860:
815:
811:
81:
1039:
4838:
4772:
4613:
3223:
1727:
442:, one obtains a stronger requirement. A function with this property is called
1272:
4889:
4762:
4568:
4200:
4009:
3631:
1251:
1402:
is continuous exactly at every irrational number (cf. picture). It is the
4684:
4551:
4302:
3681:
2855:
2045:{\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}
121:
105:
38:. For information on monotonicity as it pertains to logical systems, see
3016:
in its domain. The composite of two monotone mappings is also monotone.
3755:
3480:
3973:
Conditions for optimality: Admissibility and consistency pg. 94–95 (
145:
3420:, monotonicity is a stricter requirement than admissibility. Some
307:
preserves the order (see Figure 1). Likewise, a function is called
3999:
3987:
Bayless, Sam; Bayless, Noah; Hoos, Holger H.; Hu, Alan J. (2015).
3479:
3443:
2930:
denote the partial order relation of any partially ordered set, a
422:
in the definition of monotonicity is replaced by the strict order
67:
59:
51:
3630:-ary Boolean function is monotonic when its representation as an
3238:
monotonicity (also called consistency) is a condition applied to
727:
increasing, but it is neither non-decreasing nor non-increasing.
1840:. In addition, this result cannot be improved to countable: see
4213:
4209:
4121:(3rd ed.). Upper Saddle River, New Jersey: Prentice Hall.
3384:{\displaystyle h(n)\leq c\left(n,a,n'\right)+h\left(n'\right).}
108:, and was later generalized to the more abstract setting of
3804:
Absolutely and completely monotonic functions and sequences
1047:
The following properties are true for a monotonic function
3484:
Hasse diagram of the monotonic function "at least two of
1659:. Other important properties of these functions include:
4043:
Lattice theory: first concepts and distributive lattices
3944:(8th ed.). W. W. Norton & Company. p. 56.
2512:{\displaystyle (Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.}
2061:
if it is monotonically increasing up to some point (the
3906:. Houston, Texas: Publish or Perish, Inc. p. 192.
1110:
from the right and from the left at every point of its
1019:
being preserved across a monotonic transform (see also
701:{\displaystyle f\!\left(x\right)\neq f\!\left(y\right)}
388:{\displaystyle f\!\left(x\right)\geq f\!\left(y\right)}
280:{\displaystyle f\!\left(x\right)\leq f\!\left(y\right)}
4183:
by Anik
Debnath and Thomas Roxlo (The Harker School),
3926:
See the section on
Cardinal Versus Ordinal Utility in
1954:
An important application of monotonic functions is in
651:{\displaystyle f\!\left(x\right)>f\!\left(y\right)}
601:{\displaystyle f\!\left(x\right)<f\!\left(y\right)}
3307:
3248:
3177:
3149:
3047:
2954:
2916:
2893:
2873:
2834:
is a monotone set. A monotone operator is said to be
2811:
2787:
2710:
2688:
2642:
2596:
2559:
2539:
2447:
2402:
2382:
2345:
2302:
2273:
2229:
2192:
2172:
2148:
2124:
2100:
2076:
1991:
1964:
1932:
1896:
1876:
1853:
1819:
1799:
1779:
1759:
1739:
1712:
1692:
1669:
1621:
1588:
1534:
1497:
1473:
1443:
1416:
1343:
1306:
1236:
1210:
1185:
1165:
1142:
1122:
1092:
1053:
944:
909:
877:
842:
796:
760:
736:
664:
614:
564:
538:
512:
492:
472:
428:
408:
351:
325:
293:
243:
217:
197:
177:
130:
4062:
Mathematics for economists: an introductory textbook
3123:
is both monotone and antitone, and if the domain of
1647:, then its derivative is positive at every point in
1076:{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }
723:
are often used to refer to non-strict monotonicity.
4755:
4683:
4622:
4392:
4321:
4270:
3637:labelled with truth values has no upward edge from
3432:provided that the heuristic they use is monotonic.
4098:Riesz, Frigyes & Béla Szőkefalvi-Nagy (1990).
3383:
3295:plus the estimated cost of reaching the goal from
3263:
3210:
3161:
3094:
2998:
2922:
2899:
2879:
2826:
2793:
2774:
2694:
2674:
2628:
2578:
2545:
2511:
2427:
2388:
2354:
2327:
2288:
2247:
2198:
2178:
2154:
2130:
2106:
2082:
2044:
1970:
1938:
1918:
1882:
1859:
1825:
1805:
1785:
1765:
1745:
1718:
1698:
1675:
1627:
1607:
1567:
1520:
1479:
1456:
1429:
1394:
1325:
1292:
1242:
1216:
1194:
1171:
1151:
1128:
1098:
1075:
980:
930:
895:
863:
802:
782:
742:
700:
650:
600:
550:
524:
498:
478:
434:
414:
387:
337:
299:
279:
229:
203:
183:
136:
4081:An introduction to partial differential equations
4079:Renardy, Michael & Rogers, Robert C. (2004).
2022:
2002:
1643:is differentiable and increasing on an interval,
686:
668:
636:
618:
586:
568:
373:
355:
265:
247:
64:Figure 2. A monotonically non-increasing function
56:Figure 1. A monotonically non-decreasing function
3508:, a monotonic function is one such that for all
2775:{\displaystyle (w_{1}-w_{2},u_{1}-u_{2})\geq 0.}
45:"Monotonic" redirects here. For other uses, see
3823:Clapham, Christopher; Nicholson, James (2014).
3291:is no greater than the step cost of getting to
3287:, the estimated cost of reaching the goal from
2530:have monotonic operators as their derivatives.
1890:is a monotonic function defined on an interval
4136:Simon, Carl P.; Blume, Lawrence (April 1994).
3657:, which is the more common representation for
3496:hold". Colors indicate function output values.
3119:is both monotone and antitone; conversely, if
3095:{\displaystyle x\leq y\implies f(y)\leq f(x),}
1136:has a limit at positive or negative infinity (
4225:
3974:
3685:is forbidden). For instance "at least two of
3416:. Because every monotonic heuristic is also
2999:{\displaystyle x\leq y\implies f(x)\leq f(y)}
1395:{\displaystyle f(x)=\sum _{q_{i}\leq x}a_{i}}
458:). A function with either property is called
8:
3709:, since it can be written for instance as ((
462:. Functions that are strictly monotone are
4883:Positive cone of a partially ordered group
4232:
4218:
4210:
4119:Artificial Intelligence: A Modern Approach
4117:Russell, Stuart J.; Norvig, Peter (2010).
4060:Pemberton, Malcolm; Rau, Nicholas (2001).
3927:
3780:- measure of monotonicity in a set of data
3061:
3057:
2968:
2964:
4008:
3998:
3306:
3247:
3176:
3148:
3046:
2953:
2915:
2892:
2872:
2810:
2786:
2757:
2744:
2731:
2718:
2709:
2687:
2663:
2650:
2641:
2617:
2604:
2595:
2570:
2558:
2538:
2446:
2419:
2401:
2381:
2344:
2307:
2301:
2272:
2228:
2191:
2171:
2147:
2123:
2099:
2075:
2017:
1996:
1990:
1963:
1931:
1895:
1875:
1852:
1818:
1798:
1778:
1758:
1738:
1711:
1691:
1668:
1620:
1593:
1587:
1550:
1533:
1513:
1512:
1511:
1502:
1496:
1472:
1448:
1442:
1421:
1415:
1386:
1368:
1363:
1342:
1314:
1305:
1282:
1235:
1209:
1184:
1164:
1141:
1121:
1091:
1069:
1068:
1061:
1060:
1052:
943:
908:
876:
841:
795:
759:
735:
663:
613:
563:
537:
511:
491:
471:
427:
407:
350:
324:
292:
242:
216:
196:
176:
129:
4866:Positive cone of an ordered vector space
3827:(5th ed.). Oxford University Press.
3825:Oxford Concise Dictionary of Mathematics
2052:is a monotonically increasing function.
1337:, the monotonically increasing function
1300:of positive numbers and any enumeration
1038:
1030:
3964:if its domain has more than one element
3815:
3778:Spearman's rank correlation coefficient
1683:is a monotonic function defined on an
1521:{\displaystyle x^{*}\in {\mathbb {R}}}
27:Order-preserving mathematical function
2067:) and then monotonically decreasing.
1043:Plots of 6 monotonic growth functions
826:All strictly monotonic functions are
100:that preserves or reverses the given
7:
3897:
3895:
3870:
3868:
3866:
3864:
3862:
3836:
3834:
2428:{\displaystyle T:X\rightarrow X^{*}}
871:is strictly increasing on the range
790:if the derivatives of all orders of
4181:Convergence of a Monotonic Sequence
3448:With the nonmonotonic function "if
3230:In the context of search algorithms
2396:, a (possibly non-linear) operator
1027:Some basic applications and results
4393:Properties & Types (
4017:from the original on Dec 11, 2023.
2850:Order theory deals with arbitrary
2488:
1189:
1166:
1146:
25:
4849:Positive cone of an ordered field
3697:hold" is a monotonic function of
1575:, then there is a non-degenerate
4703:Ordered topological vector space
3736:The number of such functions on
3271:is monotonic if, for every node
1984:cumulative distribution function
1404:cumulative distribution function
783:{\displaystyle \left(a,b\right)}
558:and so, by monotonicity, either
4064:. Manchester University Press.
2487:
1847:if this set is countable, then
1410:on the rational numbers, where
818:at all points on the interval.
4185:Wolfram Demonstrations Project
4102:. Courier Dover Publications.
3317:
3311:
3258:
3252:
3211:{\displaystyle f(x)\leq f(y))}
3205:
3202:
3196:
3187:
3181:
3086:
3080:
3071:
3065:
3058:
3035:. Hence, an antitone function
2993:
2987:
2978:
2972:
2965:
2821:
2815:
2763:
2711:
2669:
2643:
2623:
2597:
2478:
2448:
2412:
2322:
2316:
2239:
1568:{\displaystyle f'(x^{*})>0}
1556:
1543:
1353:
1347:
1320:
1307:
1288:
1275:
1065:
995:is sometimes used in place of
975:
972:
966:
957:
951:
945:
925:
919:
890:
878:
858:
852:
711:To avoid ambiguity, the terms
104:. This concept first arose in
1:
4660:Series-parallel partial order
4034:The elements of real analysis
3990:SAT Modulo Monotonic Theories
2579:{\displaystyle X\times X^{*}}
1639:. As a partial converse, if
72:Figure 3. A function that is
4339:Cantor's isomorphism theorem
3768:Monotone cubic interpolation
3582:(i.e. the Cartesian product
2267:; that is, for each element
152:with real values is called
4379:Szpilrajn extension theorem
4354:Hausdorff maximal principle
4329:Boolean prime ideal theorem
4169:Encyclopedia of Mathematics
3942:Intermediate Microeconomics
3880:Encyclopedia of Mathematics
3653:of the function's labelled
1159:) of either a real number,
1152:{\displaystyle \pm \infty }
4947:
4725:Topological vector lattice
4138:Mathematics for Economists
4032:Bartle, Robert G. (1976).
3794:Operator monotone function
3740:variables is known as the
1753:; i.e. the set of numbers
1608:{\displaystyle x^{*}\in I}
399:the order (see Figure 2).
44:
40:Monotonicity of entailment
29:
4247:
3975:Russell & Norvig 2010
2946:, satisfies the property
2328:{\displaystyle f^{-1}(y)}
2296:the (possibly empty) set
1813:is not differentiable in
903:, then it has an inverse
47:Monotone (disambiguation)
4334:Cantor–Bernstein theorem
4041:Grätzer, George (1971).
3928:Simon & Blume (1994)
3902:Spivak, Michael (1994).
3773:Pseudo-monotone operator
3283:generated by any action
2375:topological vector space
2248:{\displaystyle f:X\to Y}
1867:is absolutely continuous
1266:). For example, for any
1195:{\displaystyle -\infty }
1009:monotonic transformation
1003:Monotonic transformation
309:monotonically decreasing
161:monotonically increasing
116:In calculus and analysis
4878:Partially ordered group
4698:Specialization preorder
4010:10.1609/aaai.v29i1.9755
3940:Varian, Hal R. (2010).
3162:{\displaystyle x\leq y}
3039:satisfies the property
3023:notion is often called
2289:{\displaystyle y\in Y,}
1326:{\displaystyle (q_{i})}
1172:{\displaystyle \infty }
1013:monotone transformation
338:{\displaystyle x\leq y}
230:{\displaystyle x\leq y}
4364:Kruskal's tree theorem
4359:Knaster–Tarski theorem
4349:Dushnik–Miller theorem
3497:
3469:
3385:
3265:
3212:
3163:
3096:
3000:
2934:function, also called
2924:
2901:
2881:
2852:partially ordered sets
2828:
2795:
2776:
2696:
2676:
2630:
2580:
2547:
2520:Kachurovskii's theorem
2513:
2429:
2390:
2365:In functional analysis
2356:
2329:
2290:
2249:
2200:
2180:
2156:
2132:
2118:on its domain, and if
2108:
2084:
2046:
1972:
1940:
1920:
1884:
1861:
1827:
1807:
1787:
1767:
1747:
1720:
1700:
1677:
1629:
1609:
1569:
1522:
1481:
1458:
1431:
1396:
1327:
1294:
1244:
1218:
1196:
1173:
1153:
1130:
1100:
1077:
1044:
1036:
982:
932:
931:{\displaystyle x=h(y)}
897:
865:
864:{\displaystyle y=g(x)}
804:
784:
744:
702:
652:
602:
552:
551:{\displaystyle x>y}
526:
525:{\displaystyle x<y}
500:
480:
436:
416:
389:
339:
301:
281:
231:
205:
185:
138:
77:
65:
57:
36:monotonicity criterion
3841:Stover, Christopher.
3799:Monotone set function
3789:Cyclical monotonicity
3483:
3447:
3386:
3266:
3213:
3164:
3143:(functions for which
3097:
3001:
2925:
2923:{\displaystyle \leq }
2902:
2882:
2829:
2796:
2777:
2697:
2677:
2631:
2581:
2548:
2514:
2430:
2391:
2357:
2330:
2291:
2250:
2201:
2181:
2157:
2133:
2109:
2085:
2047:
1973:
1941:
1921:
1919:{\displaystyle \left}
1885:
1862:
1828:
1808:
1788:
1768:
1748:
1721:
1701:
1678:
1630:
1610:
1570:
1523:
1482:
1459:
1457:{\displaystyle q_{i}}
1432:
1430:{\displaystyle a_{i}}
1397:
1328:
1295:
1245:
1219:
1197:
1174:
1154:
1131:
1101:
1078:
1042:
1034:
983:
933:
898:
866:
805:
785:
745:
703:
653:
603:
553:
527:
501:
481:
437:
417:
415:{\displaystyle \leq }
390:
340:
302:
282:
232:
206:
186:
159:A function is termed
139:
71:
63:
55:
4856:Ordered vector space
4196:"Monotonic Function"
3843:"Monotonic Function"
3436:In Boolean functions
3422:heuristic algorithms
3305:
3275:and every successor
3264:{\displaystyle h(n)}
3246:
3175:
3147:
3045:
2952:
2914:
2900:{\displaystyle >}
2891:
2880:{\displaystyle <}
2871:
2840:maximal monotone set
2827:{\displaystyle G(T)}
2809:
2785:
2708:
2686:
2640:
2594:
2557:
2537:
2445:
2400:
2380:
2343:
2300:
2271:
2227:
2190:
2170:
2146:
2122:
2098:
2074:
1989:
1962:
1930:
1894:
1874:
1851:
1817:
1797:
1777:
1757:
1737:
1710:
1690:
1667:
1619:
1586:
1532:
1495:
1471:
1441:
1414:
1341:
1304:
1273:
1234:
1226:jump discontinuities
1208:
1183:
1163:
1140:
1120:
1090:
1051:
1021:monotone preferences
942:
907:
875:
840:
794:
758:
752:absolutely monotonic
734:
662:
612:
562:
536:
510:
490:
470:
435:{\displaystyle <}
426:
406:
349:
323:
291:
241:
215:
195:
175:
128:
4916:Functional analysis
4694:Alexandrov topology
4640:Lexicographic order
4599:Well-quasi-ordering
4164:"Monotone function"
4100:Functional Analysis
3876:"Monotone function"
3464:nodes appear above
3395:triangle inequality
3240:heuristic functions
3226:order embeddings).
2863:. Furthermore, the
2371:functional analysis
2162:, then there is an
822:Inverse of function
452:strictly decreasing
444:strictly increasing
18:Increasing function
4931:Types of functions
4675:Transitive closure
4635:Converse/Transpose
4344:Dilworth's theorem
4193:Weisstein, Eric W.
4140:(first ed.).
4036:(second ed.).
3784:Total monotonicity
3645:. (This labelled
3498:
3470:
3393:This is a form of
3381:
3261:
3234:In the context of
3220:order isomorphisms
3208:
3159:
3135:must be constant.
3092:
2996:
2920:
2897:
2877:
2838:if its graph is a
2824:
2791:
2772:
2692:
2672:
2626:
2590:if for every pair
2576:
2543:
2509:
2425:
2386:
2355:{\displaystyle X.}
2352:
2325:
2286:
2245:
2196:
2176:
2152:
2128:
2104:
2092:strictly monotonic
2080:
2042:
1968:
1956:probability theory
1948:Riemann integrable
1936:
1916:
1880:
1857:
1823:
1803:
1783:
1763:
1743:
1716:
1696:
1673:
1625:
1605:
1565:
1518:
1477:
1454:
1427:
1392:
1381:
1323:
1240:
1214:
1192:
1169:
1149:
1126:
1096:
1073:
1045:
1037:
997:strictly monotonic
978:
928:
893:
861:
800:
780:
740:
698:
648:
598:
548:
522:
496:
476:
432:
412:
385:
335:
297:
277:
227:
201:
181:
134:
86:monotonic function
78:
66:
58:
4903:
4902:
4861:Partially ordered
4670:Symmetric closure
4655:Reflexive closure
4398:
4152:(Definition 9.31)
4147:978-0-393-95733-4
4128:978-0-13-604259-4
4109:978-0-486-66289-3
3847:Wolfram MathWorld
3502:
3501:
3474:
3473:
3236:search algorithms
3117:constant function
2794:{\displaystyle G}
2695:{\displaystyle G}
2546:{\displaystyle G}
2437:monotone operator
2389:{\displaystyle X}
2199:{\displaystyle f}
2179:{\displaystyle T}
2155:{\displaystyle f}
2131:{\displaystyle T}
2107:{\displaystyle f}
2083:{\displaystyle f}
2020:
1971:{\displaystyle X}
1939:{\displaystyle f}
1883:{\displaystyle f}
1860:{\displaystyle f}
1826:{\displaystyle x}
1806:{\displaystyle f}
1786:{\displaystyle I}
1766:{\displaystyle x}
1746:{\displaystyle I}
1731:almost everywhere
1719:{\displaystyle f}
1699:{\displaystyle I}
1676:{\displaystyle f}
1635:is increasing on
1628:{\displaystyle f}
1480:{\displaystyle f}
1437:is the weight of
1359:
1268:summable sequence
1243:{\displaystyle f}
1217:{\displaystyle f}
1129:{\displaystyle f}
1099:{\displaystyle f}
803:{\displaystyle f}
754:over an interval
743:{\displaystyle f}
721:weakly decreasing
717:weakly increasing
499:{\displaystyle y}
479:{\displaystyle x}
460:strictly monotone
300:{\displaystyle f}
204:{\displaystyle y}
184:{\displaystyle x}
137:{\displaystyle f}
90:monotone function
16:(Redirected from
4938:
4645:Linear extension
4394:
4374:Mirsky's theorem
4234:
4227:
4220:
4211:
4206:
4205:
4177:
4151:
4132:
4113:
4094:
4075:
4056:
4037:
4019:
4018:
4012:
4002:
3984:
3978:
3971:
3965:
3962:
3956:
3955:
3937:
3931:
3924:
3918:
3917:
3899:
3890:
3889:
3887:
3886:
3872:
3857:
3856:
3854:
3853:
3838:
3829:
3828:
3820:
3747:
3739:
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3688:
3663:
3634:
3629:
3625:
3585:
3581:
3561:
3545:
3529:
3525:
3516:
3495:
3491:
3487:
3476:
3475:
3467:
3463:
3459:
3455:
3451:
3440:
3439:
3415:
3411:
3404:
3400:
3390:
3388:
3387:
3382:
3377:
3373:
3355:
3351:
3350:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3268:
3267:
3262:
3217:
3215:
3214:
3209:
3168:
3166:
3165:
3160:
3141:order embeddings
3134:
3126:
3122:
3111:
3107:
3101:
3099:
3098:
3093:
3038:
3015:
3011:
3005:
3003:
3002:
2997:
2944:
2943:
2942:order-preserving
2929:
2927:
2926:
2921:
2906:
2904:
2903:
2898:
2886:
2884:
2883:
2878:
2836:maximal monotone
2833:
2831:
2830:
2825:
2803:maximal monotone
2800:
2798:
2797:
2792:
2781:
2779:
2778:
2773:
2762:
2761:
2749:
2748:
2736:
2735:
2723:
2722:
2701:
2699:
2698:
2693:
2681:
2679:
2678:
2675:{\displaystyle }
2673:
2668:
2667:
2655:
2654:
2635:
2633:
2632:
2629:{\displaystyle }
2627:
2622:
2621:
2609:
2608:
2586:is said to be a
2585:
2583:
2582:
2577:
2575:
2574:
2552:
2550:
2549:
2544:
2524:convex functions
2518:
2516:
2515:
2510:
2435:is said to be a
2434:
2432:
2431:
2426:
2424:
2423:
2395:
2393:
2392:
2387:
2361:
2359:
2358:
2353:
2334:
2332:
2331:
2326:
2315:
2314:
2295:
2293:
2292:
2287:
2254:
2252:
2251:
2246:
2205:
2203:
2202:
2197:
2185:
2183:
2182:
2177:
2164:inverse function
2161:
2159:
2158:
2153:
2137:
2135:
2134:
2129:
2113:
2111:
2110:
2105:
2089:
2087:
2086:
2081:
2051:
2049:
2048:
2043:
2041:
2037:
2021:
2018:
2013:
2001:
2000:
1977:
1975:
1974:
1969:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1915:
1911:
1889:
1887:
1886:
1881:
1866:
1864:
1863:
1858:
1832:
1830:
1829:
1824:
1812:
1810:
1809:
1804:
1792:
1790:
1789:
1784:
1772:
1770:
1769:
1764:
1752:
1750:
1749:
1744:
1725:
1723:
1722:
1717:
1705:
1703:
1702:
1697:
1682:
1680:
1679:
1674:
1634:
1632:
1631:
1626:
1614:
1612:
1611:
1606:
1598:
1597:
1574:
1572:
1571:
1566:
1555:
1554:
1542:
1527:
1525:
1524:
1519:
1517:
1516:
1507:
1506:
1486:
1484:
1483:
1478:
1463:
1461:
1460:
1455:
1453:
1452:
1436:
1434:
1433:
1428:
1426:
1425:
1408:discrete measure
1401:
1399:
1398:
1393:
1391:
1390:
1380:
1373:
1372:
1335:rational numbers
1332:
1330:
1329:
1324:
1319:
1318:
1299:
1297:
1296:
1291:
1287:
1286:
1249:
1247:
1246:
1241:
1223:
1221:
1220:
1215:
1201:
1199:
1198:
1193:
1178:
1176:
1175:
1170:
1158:
1156:
1155:
1150:
1135:
1133:
1132:
1127:
1105:
1103:
1102:
1097:
1082:
1080:
1079:
1074:
1072:
1064:
1017:utility function
987:
985:
984:
981:{\displaystyle }
979:
937:
935:
934:
929:
902:
900:
899:
896:{\displaystyle }
894:
870:
868:
867:
862:
809:
807:
806:
801:
789:
787:
786:
781:
779:
775:
749:
747:
746:
741:
707:
705:
704:
699:
697:
679:
657:
655:
654:
649:
647:
629:
607:
605:
604:
599:
597:
579:
557:
555:
554:
549:
531:
529:
528:
523:
505:
503:
502:
497:
485:
483:
482:
477:
441:
439:
438:
433:
421:
419:
418:
413:
394:
392:
391:
386:
384:
366:
344:
342:
341:
336:
306:
304:
303:
298:
286:
284:
283:
278:
276:
258:
236:
234:
233:
228:
210:
208:
207:
202:
190:
188:
187:
182:
143:
141:
140:
135:
21:
4946:
4945:
4941:
4940:
4939:
4937:
4936:
4935:
4906:
4905:
4904:
4899:
4895:Young's lattice
4751:
4679:
4618:
4468:Heyting algebra
4416:Boolean algebra
4388:
4369:Laver's theorem
4317:
4283:Boolean algebra
4278:Binary relation
4266:
4243:
4238:
4191:
4190:
4162:
4159:
4148:
4135:
4129:
4116:
4110:
4097:
4091:
4078:
4072:
4059:
4053:
4040:
4031:
4028:
4023:
4022:
3986:
3985:
3981:
3972:
3968:
3963:
3959:
3952:
3939:
3938:
3934:
3925:
3921:
3914:
3901:
3900:
3893:
3884:
3882:
3874:
3873:
3860:
3851:
3849:
3840:
3839:
3832:
3822:
3821:
3817:
3812:
3764:
3754:, generally an
3745:
3742:Dedekind number
3737:
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3679:(in particular
3658:
3632:
3627:
3623:
3614:
3607:
3598:
3591:
3583:
3580:
3571:
3563:
3560:
3553:
3547:
3544:
3537:
3531:
3527:
3524:
3518:
3515:
3509:
3506:Boolean algebra
3493:
3489:
3485:
3465:
3461:
3457:
3453:
3449:
3438:
3413:
3410:
3406:
3405:, and the goal
3402:
3398:
3366:
3362:
3343:
3330:
3326:
3303:
3302:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3244:
3243:
3242:. A heuristic
3232:
3173:
3172:
3145:
3144:
3132:
3124:
3120:
3112:in its domain.
3109:
3105:
3043:
3042:
3036:
3033:order-reversing
3013:
3009:
2950:
2949:
2941:
2940:
2912:
2911:
2889:
2888:
2869:
2868:
2856:preordered sets
2848:
2846:In order theory
2807:
2806:
2783:
2782:
2753:
2740:
2727:
2714:
2706:
2705:
2684:
2683:
2659:
2646:
2638:
2637:
2613:
2600:
2592:
2591:
2566:
2555:
2554:
2535:
2534:
2443:
2442:
2415:
2398:
2397:
2378:
2377:
2367:
2341:
2340:
2335:is a connected
2303:
2298:
2297:
2269:
2268:
2259:if each of its
2225:
2224:
2219:
2188:
2187:
2168:
2167:
2144:
2143:
2120:
2119:
2096:
2095:
2094:function, then
2072:
2071:
2027:
2023:
2003:
1992:
1987:
1986:
1980:random variable
1960:
1959:
1928:
1927:
1901:
1897:
1892:
1891:
1872:
1871:
1849:
1848:
1842:Cantor function
1815:
1814:
1795:
1794:
1775:
1774:
1755:
1754:
1735:
1734:
1708:
1707:
1688:
1687:
1665:
1664:
1617:
1616:
1589:
1584:
1583:
1546:
1535:
1530:
1529:
1498:
1493:
1492:
1469:
1468:
1444:
1439:
1438:
1417:
1412:
1411:
1382:
1364:
1339:
1338:
1310:
1302:
1301:
1278:
1274:
1271:
1270:
1256:discontinuities
1232:
1231:
1206:
1205:
1181:
1180:
1161:
1160:
1138:
1137:
1118:
1117:
1088:
1087:
1049:
1048:
1029:
1005:
940:
939:
905:
904:
873:
872:
838:
837:
824:
792:
791:
765:
761:
756:
755:
732:
731:
713:weakly monotone
687:
669:
660:
659:
637:
619:
610:
609:
587:
569:
560:
559:
534:
533:
508:
507:
488:
487:
468:
467:
424:
423:
404:
403:
374:
356:
347:
346:
321:
320:
319:) if, whenever
289:
288:
266:
248:
239:
238:
213:
212:
193:
192:
173:
172:
126:
125:
118:
50:
43:
28:
23:
22:
15:
12:
11:
5:
4944:
4942:
4934:
4933:
4928:
4923:
4918:
4908:
4907:
4901:
4900:
4898:
4897:
4892:
4887:
4886:
4885:
4875:
4874:
4873:
4868:
4863:
4853:
4852:
4851:
4841:
4836:
4835:
4834:
4829:
4822:Order morphism
4819:
4818:
4817:
4807:
4802:
4797:
4792:
4787:
4786:
4785:
4775:
4770:
4765:
4759:
4757:
4753:
4752:
4750:
4749:
4748:
4747:
4742:
4740:Locally convex
4737:
4732:
4722:
4720:Order topology
4717:
4716:
4715:
4713:Order topology
4710:
4700:
4690:
4688:
4681:
4680:
4678:
4677:
4672:
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4626:
4624:
4620:
4619:
4617:
4616:
4606:
4596:
4591:
4586:
4581:
4576:
4571:
4566:
4561:
4560:
4559:
4549:
4544:
4543:
4542:
4537:
4532:
4527:
4525:Chain-complete
4517:
4512:
4511:
4510:
4505:
4500:
4495:
4490:
4480:
4475:
4470:
4465:
4460:
4450:
4445:
4440:
4435:
4430:
4425:
4424:
4423:
4413:
4408:
4402:
4400:
4390:
4389:
4387:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4325:
4323:
4319:
4318:
4316:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4274:
4272:
4268:
4267:
4265:
4264:
4259:
4254:
4248:
4245:
4244:
4239:
4237:
4236:
4229:
4222:
4214:
4208:
4207:
4188:
4178:
4158:
4157:External links
4155:
4154:
4153:
4146:
4133:
4127:
4114:
4108:
4095:
4089:
4076:
4070:
4057:
4051:
4038:
4027:
4024:
4021:
4020:
3979:
3966:
3957:
3950:
3932:
3919:
3912:
3891:
3858:
3830:
3814:
3813:
3811:
3808:
3807:
3806:
3801:
3796:
3791:
3786:
3781:
3775:
3770:
3763:
3760:
3619:
3612:
3603:
3596:
3588:coordinatewise
3576:
3567:
3558:
3551:
3542:
3535:
3520:
3511:
3500:
3499:
3472:
3471:
3437:
3434:
3428:can be proven
3408:
3380:
3376:
3372:
3369:
3365:
3361:
3358:
3354:
3349:
3346:
3342:
3339:
3336:
3333:
3329:
3325:
3322:
3319:
3316:
3313:
3310:
3260:
3257:
3254:
3251:
3231:
3228:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3170:if and only if
3158:
3155:
3152:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3060:
3056:
3053:
3050:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2967:
2963:
2960:
2957:
2919:
2896:
2876:
2847:
2844:
2823:
2820:
2817:
2814:
2801:is said to be
2790:
2771:
2768:
2765:
2760:
2756:
2752:
2747:
2743:
2739:
2734:
2730:
2726:
2721:
2717:
2713:
2691:
2671:
2666:
2662:
2658:
2653:
2649:
2645:
2625:
2620:
2616:
2612:
2607:
2603:
2599:
2573:
2569:
2565:
2562:
2542:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2422:
2418:
2414:
2411:
2408:
2405:
2385:
2366:
2363:
2351:
2348:
2324:
2321:
2318:
2313:
2310:
2306:
2285:
2282:
2279:
2276:
2255:is said to be
2244:
2241:
2238:
2235:
2232:
2218:
2215:
2195:
2175:
2151:
2127:
2103:
2079:
2055:A function is
2040:
2036:
2033:
2030:
2026:
2016:
2012:
2009:
2006:
1999:
1995:
1967:
1952:
1951:
1935:
1914:
1910:
1907:
1904:
1900:
1879:
1868:
1856:
1845:
1822:
1802:
1782:
1762:
1742:
1728:differentiable
1715:
1695:
1672:
1653:
1652:
1624:
1604:
1601:
1596:
1592:
1564:
1561:
1558:
1553:
1549:
1545:
1541:
1538:
1515:
1510:
1505:
1501:
1489:differentiable
1476:
1465:
1451:
1447:
1424:
1420:
1389:
1385:
1379:
1376:
1371:
1367:
1362:
1358:
1355:
1352:
1349:
1346:
1322:
1317:
1313:
1309:
1290:
1285:
1281:
1277:
1250:can only have
1239:
1229:
1224:can only have
1213:
1203:
1191:
1188:
1168:
1148:
1145:
1125:
1115:
1095:
1071:
1067:
1063:
1059:
1056:
1028:
1025:
1004:
1001:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
927:
924:
921:
918:
915:
912:
892:
889:
886:
883:
880:
860:
857:
854:
851:
848:
845:
823:
820:
799:
778:
774:
771:
768:
764:
750:is said to be
739:
696:
693:
690:
685:
682:
678:
675:
672:
667:
646:
643:
640:
635:
632:
628:
625:
622:
617:
596:
593:
590:
585:
582:
578:
575:
572:
567:
547:
544:
541:
521:
518:
515:
495:
475:
431:
411:
383:
380:
377:
372:
369:
365:
362:
359:
354:
334:
331:
328:
317:non-increasing
296:
275:
272:
269:
264:
261:
257:
254:
251:
246:
226:
223:
220:
200:
180:
169:non-decreasing
133:
117:
114:
32:voting systems
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4943:
4932:
4929:
4927:
4926:Real analysis
4924:
4922:
4919:
4917:
4914:
4913:
4911:
4896:
4893:
4891:
4888:
4884:
4881:
4880:
4879:
4876:
4872:
4869:
4867:
4864:
4862:
4859:
4858:
4857:
4854:
4850:
4847:
4846:
4845:
4844:Ordered field
4842:
4840:
4837:
4833:
4830:
4828:
4825:
4824:
4823:
4820:
4816:
4813:
4812:
4811:
4808:
4806:
4803:
4801:
4800:Hasse diagram
4798:
4796:
4793:
4791:
4788:
4784:
4781:
4780:
4779:
4778:Comparability
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4760:
4758:
4754:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4727:
4726:
4723:
4721:
4718:
4714:
4711:
4709:
4706:
4705:
4704:
4701:
4699:
4695:
4692:
4691:
4689:
4686:
4682:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4650:Product order
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4623:Constructions
4621:
4615:
4611:
4607:
4604:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4558:
4555:
4554:
4553:
4550:
4548:
4545:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4522:
4521:
4520:Partial order
4518:
4516:
4513:
4509:
4508:Join and meet
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4485:
4484:
4481:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4455:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4422:
4419:
4418:
4417:
4414:
4412:
4409:
4407:
4406:Antisymmetric
4404:
4403:
4401:
4397:
4391:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4326:
4324:
4320:
4314:
4313:Weak ordering
4311:
4309:
4306:
4304:
4301:
4299:
4298:Partial order
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4275:
4273:
4269:
4263:
4260:
4258:
4255:
4253:
4250:
4249:
4246:
4242:
4235:
4230:
4228:
4223:
4221:
4216:
4215:
4212:
4203:
4202:
4197:
4194:
4189:
4186:
4182:
4179:
4175:
4171:
4170:
4165:
4161:
4160:
4156:
4149:
4143:
4139:
4134:
4130:
4124:
4120:
4115:
4111:
4105:
4101:
4096:
4092:
4090:0-387-00444-0
4086:
4082:
4077:
4073:
4071:0-7190-3341-1
4067:
4063:
4058:
4054:
4052:0-7167-0442-0
4048:
4044:
4039:
4035:
4030:
4029:
4025:
4016:
4011:
4006:
4001:
3996:
3992:
3991:
3983:
3980:
3976:
3970:
3967:
3961:
3958:
3953:
3951:9780393934243
3947:
3943:
3936:
3933:
3929:
3923:
3920:
3915:
3913:0-914098-89-6
3909:
3905:
3898:
3896:
3892:
3881:
3877:
3871:
3869:
3867:
3865:
3863:
3859:
3848:
3844:
3837:
3835:
3831:
3826:
3819:
3816:
3809:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3779:
3776:
3774:
3771:
3769:
3766:
3765:
3761:
3759:
3757:
3753:
3749:
3743:
3734:
3684:
3683:
3678:
3677:
3672:
3671:
3665:
3661:
3656:
3652:
3648:
3647:Hasse diagram
3644:
3640:
3636:
3622:
3618:
3611:
3606:
3602:
3595:
3589:
3579:
3575:
3570:
3566:
3557:
3550:
3541:
3534:
3523:
3514:
3507:
3482:
3478:
3477:
3446:
3442:
3441:
3435:
3433:
3431:
3427:
3423:
3419:
3396:
3391:
3378:
3374:
3370:
3367:
3363:
3359:
3356:
3352:
3347:
3344:
3340:
3337:
3334:
3331:
3327:
3323:
3320:
3314:
3308:
3300:
3255:
3249:
3241:
3237:
3229:
3227:
3225:
3221:
3199:
3193:
3190:
3184:
3178:
3171:
3156:
3153:
3150:
3142:
3136:
3130:
3118:
3113:
3102:
3089:
3083:
3077:
3074:
3068:
3062:
3054:
3051:
3048:
3040:
3034:
3030:
3029:anti-monotone
3026:
3022:
3017:
3006:
2990:
2984:
2981:
2975:
2969:
2961:
2958:
2955:
2947:
2945:
2937:
2933:
2917:
2908:
2894:
2874:
2866:
2862:
2857:
2853:
2845:
2843:
2841:
2837:
2818:
2812:
2804:
2788:
2769:
2766:
2758:
2754:
2750:
2745:
2741:
2737:
2732:
2728:
2724:
2719:
2715:
2703:
2689:
2664:
2660:
2656:
2651:
2647:
2618:
2614:
2610:
2605:
2601:
2589:
2571:
2567:
2563:
2560:
2540:
2531:
2529:
2528:Banach spaces
2525:
2521:
2506:
2503:
2500:
2497:
2494:
2491:
2484:
2481:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2440:
2438:
2420:
2416:
2409:
2406:
2403:
2383:
2376:
2372:
2364:
2362:
2349:
2346:
2338:
2319:
2311:
2308:
2304:
2283:
2280:
2277:
2274:
2266:
2262:
2258:
2242:
2236:
2233:
2230:
2221:
2216:
2214:
2212:
2207:
2193:
2173:
2165:
2149:
2141:
2125:
2117:
2101:
2093:
2077:
2068:
2066:
2065:
2060:
2059:
2053:
2038:
2034:
2031:
2028:
2024:
2014:
2010:
2007:
2004:
1997:
1993:
1985:
1981:
1965:
1957:
1949:
1933:
1912:
1908:
1905:
1902:
1898:
1877:
1869:
1854:
1846:
1843:
1839:
1836:
1820:
1800:
1780:
1760:
1740:
1732:
1729:
1713:
1693:
1686:
1670:
1662:
1661:
1660:
1658:
1650:
1646:
1642:
1638:
1622:
1602:
1599:
1594:
1590:
1581:
1578:
1562:
1559:
1551:
1547:
1539:
1536:
1508:
1503:
1499:
1490:
1474:
1466:
1449:
1445:
1422:
1418:
1409:
1405:
1387:
1383:
1377:
1374:
1369:
1365:
1360:
1356:
1350:
1344:
1336:
1315:
1311:
1283:
1279:
1269:
1265:
1261:
1257:
1253:
1237:
1230:
1227:
1211:
1204:
1186:
1143:
1123:
1116:
1113:
1109:
1093:
1086:
1085:
1084:
1057:
1054:
1041:
1033:
1026:
1024:
1022:
1018:
1014:
1010:
1002:
1000:
998:
994:
989:
969:
963:
960:
954:
948:
938:on the range
922:
916:
913:
910:
887:
884:
881:
855:
849:
846:
843:
834:
831:
829:
821:
819:
817:
813:
797:
776:
772:
769:
766:
762:
753:
737:
728:
724:
722:
718:
714:
709:
694:
691:
688:
683:
680:
676:
673:
670:
665:
644:
641:
638:
633:
630:
626:
623:
620:
615:
594:
591:
588:
583:
580:
576:
573:
570:
565:
545:
542:
539:
519:
516:
513:
493:
486:not equal to
473:
466:(because for
465:
461:
457:
453:
449:
445:
429:
409:
402:If the order
400:
398:
381:
378:
375:
370:
367:
363:
360:
357:
352:
332:
329:
326:
318:
314:
310:
294:
273:
270:
267:
262:
259:
255:
252:
249:
244:
224:
221:
218:
198:
178:
171:) if for all
170:
166:
162:
157:
155:
151:
147:
144:defined on a
131:
124:, a function
123:
115:
113:
111:
107:
103:
99:
95:
91:
87:
83:
75:
70:
62:
54:
48:
41:
37:
33:
19:
4921:Order theory
4821:
4687:& Orders
4665:Star product
4594:Well-founded
4547:Prefix order
4503:Distributive
4493:Complemented
4463:Foundational
4428:Completeness
4384:Zorn's lemma
4288:Cyclic order
4271:Key concepts
4241:Order theory
4199:
4167:
4137:
4118:
4099:
4080:
4061:
4042:
4033:
4026:Bibliography
3989:
3982:
3969:
3960:
3941:
3935:
3922:
3903:
3883:. Retrieved
3879:
3850:. Retrieved
3846:
3824:
3818:
3750:
3735:
3680:
3674:
3668:
3666:
3659:
3655:Venn diagram
3642:
3638:
3620:
3616:
3609:
3604:
3600:
3593:
3577:
3573:
3568:
3564:
3555:
3548:
3539:
3532:
3521:
3512:
3503:
3392:
3301:
3233:
3137:
3114:
3103:
3041:
3032:
3028:
3024:
3018:
3007:
2948:
2939:
2935:
2931:
2909:
2849:
2839:
2835:
2802:
2704:
2588:monotone set
2587:
2532:
2441:
2436:
2368:
2256:
2222:
2220:
2210:
2208:
2091:
2069:
2062:
2056:
2054:
1953:
1838:measure zero
1654:
1648:
1644:
1640:
1636:
1579:
1263:
1259:
1046:
1012:
1008:
1006:
996:
992:
990:
835:
832:
825:
751:
729:
725:
720:
716:
712:
710:
459:
455:
451:
447:
443:
401:
396:
316:
312:
308:
168:
164:
160:
158:
153:
150:real numbers
119:
110:order theory
98:ordered sets
89:
85:
79:
73:
4871:Riesz space
4832:Isomorphism
4708:Normal cone
4630:Composition
4564:Semilattice
4473:Homogeneous
4458:Equivalence
4308:Total order
3752:SAT solving
3586:is ordered
3412:closest to
2522:shows that
2217:In topology
816:nonpositive
812:nonnegative
730:A function
82:mathematics
4910:Categories
4839:Order type
4773:Cofinality
4614:Well-order
4589:Transitive
4478:Idempotent
4411:Asymmetric
3885:2018-01-29
3852:2018-01-29
3452:then both
3418:admissible
3224:surjective
2867:relations
1793:such that
1582:such that
828:invertible
464:one-to-one
456:decreasing
448:increasing
313:decreasing
211:such that
165:increasing
4890:Upper set
4827:Embedding
4763:Antichain
4584:Tolerance
4574:Symmetric
4569:Semiorder
4515:Reflexive
4433:Connected
4201:MathWorld
4174:EMS Press
4000:1406.0043
3321:≤
3191:≤
3154:≤
3075:≤
3059:⟹
3052:≤
2982:≤
2966:⟹
2959:≤
2918:≤
2767:≥
2751:−
2725:−
2572:∗
2564:×
2533:A subset
2501:∈
2489:∀
2482:≥
2473:−
2458:−
2421:∗
2413:→
2309:−
2278:∈
2265:connected
2240:→
2116:injective
2032:≤
1600:∈
1595:∗
1552:∗
1509:∈
1504:∗
1375:≤
1361:∑
1252:countably
1190:∞
1187:−
1167:∞
1147:∞
1144:±
1066:→
1058::
1007:The term
993:monotonic
991:The term
681:≠
506:, either
410:≤
368:≥
330:≤
260:≤
237:one has
222:≤
154:monotonic
76:monotonic
4685:Topology
4552:Preorder
4535:Eulerian
4498:Complete
4448:Directed
4438:Covering
4303:Preorder
4262:Category
4257:Glossary
4015:Archived
3904:Calculus
3762:See also
3590:), then
3424:such as
3371:′
3348:′
3104:for all
3025:antitone
3008:for all
2932:monotone
2910:Letting
2337:subspace
2257:monotone
2058:unimodal
1835:Lebesgue
1685:interval
1657:analysis
1577:interval
1540:′
397:reverses
395:, so it
122:calculus
106:calculus
96:between
94:function
4790:Duality
4768:Cofinal
4756:Related
4735:Fréchet
4612:)
4488:Bounded
4483:Lattice
4456:)
4454:Partial
4322:Results
4293:Lattice
4176:, 2001
3756:NP-hard
3649:is the
3615:, ...,
3599:, ...,
3562:, ...,
3430:optimal
3397:, with
3131:, then
3129:lattice
2936:isotone
2213:-axis.
2138:is the
1926:, then
1706:, then
1406:of the
1333:of the
1293:(a_{i})
814:or all
658:, thus
345:, then
148:of the
92:) is a
4815:Subnet
4795:Filter
4745:Normed
4730:Banach
4696:&
4603:Better
4540:Strict
4530:Graded
4421:topics
4252:Topics
4144:
4125:
4106:
4087:
4068:
4049:
3948:
3910:
3725:) or (
3717:) or (
3608:) ≤ f(
3584:{0, 1}
3468:nodes.
2865:strict
2261:fibers
2223:A map
1982:, its
1112:domain
1108:limits
454:(also
446:(also
311:(also
163:(also
146:subset
34:, see
4805:Ideal
4783:Graph
4579:Total
4557:Total
4443:Dense
3995:arXiv
3810:Notes
3643:false
3635:-cube
3530:, if
3528:{0,1}
3462:false
3127:is a
3031:, or
2938:, or
2861:total
2373:on a
2140:range
2090:is a
2070:When
1978:is a
1958:. If
1254:many
1179:, or
287:, so
102:order
4396:list
4142:ISBN
4123:ISBN
4104:ISBN
4085:ISBN
4066:ISBN
4047:ISBN
3946:ISBN
3908:ISBN
3733:)).
3729:and
3721:and
3713:and
3673:and
3651:dual
3639:true
3517:and
3466:true
3456:and
3218:and
3108:and
3021:dual
3019:The
3012:and
2895:>
2887:and
2875:<
2854:and
2636:and
2186:for
2064:mode
2019:Prob
1833:has
1615:and
1560:>
1528:and
1106:has
1011:(or
810:are
719:and
631:>
581:<
543:>
517:<
430:<
191:and
88:(or
84:, a
4810:Net
4610:Pre
4005:doi
3744:of
3682:not
3670:and
3664:.)
3662:≤ 3
3641:to
3526:in
3504:In
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3279:of
2682:in
2553:of
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2369:In
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2166:on
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2114:is
1946:is
1870:if
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1726:is
1663:if
1491:at
1487:is
1467:If
708:.)
608:or
532:or
315:or
167:or
120:In
80:In
74:not
4912::
4198:.
4172:,
4166:,
4045:.
4013:.
4003:.
3977:).
3894:^
3878:.
3861:^
3845:.
3833:^
3748:.
3705:,
3701:,
3693:,
3689:,
3676:or
3592:f(
3572:≤
3554:≤
3546:,
3538:≤
3492:,
3488:,
3426:A*
3403:n'
3401:,
3299:,
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2842:.
2770:0.
2702:,
1262:,
1083::
988:.
715:,
112:.
4608:(
4605:)
4601:(
4452:(
4399:)
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4226:t
4219:v
4204:.
4187:.
4150:.
4131:.
4112:.
4093:.
4074:.
4055:.
4007::
3997::
3954:.
3930:.
3916:.
3888:.
3855:.
3746:n
3738:n
3731:c
3727:b
3723:c
3719:a
3715:b
3711:a
3707:c
3703:b
3699:a
3695:c
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3687:a
3660:n
3633:n
3628:n
3624:)
3621:n
3617:b
3613:1
3610:b
3605:n
3601:a
3597:1
3594:a
3578:n
3574:b
3569:n
3565:a
3559:2
3556:b
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3543:1
3540:b
3536:1
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3522:i
3519:b
3513:i
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3490:b
3486:a
3458:c
3454:b
3450:a
3414:n
3409:n
3407:G
3399:n
3379:.
3375:)
3368:n
3364:(
3360:h
3357:+
3353:)
3345:n
3341:,
3338:a
3335:,
3332:n
3328:(
3324:c
3318:)
3315:n
3312:(
3309:h
3289:n
3285:a
3281:n
3273:n
3259:)
3256:n
3253:(
3250:h
3222:(
3206:)
3203:)
3200:y
3197:(
3194:f
3188:)
3185:x
3182:(
3179:f
3157:y
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2126:T
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2039:)
2035:x
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2025:(
2015:=
2011:)
2008:x
2005:(
1998:X
1994:F
1966:X
1950:.
1934:f
1913:]
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1899:[
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1844:.
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970:b
967:(
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952:(
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920:(
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914:=
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891:]
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