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with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance:
97:, that is, operations defined on functions by applying the operations to function values separately for each point in the
17:
1606:
1566:
1176:, and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.
515:
1000:
932:
437:
1243:. Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are
113:
1137:
1200:
789:
Componentwise operations are usually defined on vectors, where vectors are elements of the set
68:
1105:
591:
588:
The pointwise product with a scalar is usually written with the scalar term first. Thus, when
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8:
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are denoted by the same symbol. A similar definition is used for unary operations
1311:
1079:
38:
is used to indicate that a certain property is defined by considering each value
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can be turned into an algebraic structure of the same type in an analogous way.
817:
706:
1082:
can be regarded as a function, and a vector is a tuple. Therefore, any vector
1600:
1189:
702:
698:
118:
997:
Componentwise operations can be defined on matrices. Matrix addition, where
1185:
27:
Applying operations to functions in terms of values for each input "point"
1573:
758:
686:
125:(red). The highlighted vertical slice shows the computation at the point
31:
117:
Pointwise sum (upper plot, violet) and product (green) of the functions
1586:
1330:
1277:
1338:
710:
429:
1572:
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove,
351:
672:{\displaystyle (\lambda \cdot f)(x)=\lambda \cdot f(x).}
510:
The pointwise product or pointwise multiplication is:
1585:
This article incorporates material from
Pointwise on
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1508:{\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x).}
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59:
681:An example of an operation on functions which is
93:An important class of pointwise concepts are the
1598:
1591:Creative Commons Attribution/Share-Alike License
1453:
697:Pointwise operations inherit such properties as
579:{\displaystyle (f\cdot g)(x)=f(x)\cdot g(x).}
784:
1559:Lattices and Ordered Algebraic Structures
1424:{\displaystyle f_{n}:X\longrightarrow Y}
1064:{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}}
160:can be lifted pointwise to an operation
112:
1385:{\displaystyle (f_{n})_{n=1}^{\infty }}
108:
14:
1599:
1288:) with the additional property that id
1179:
987:{\displaystyle (u+v)_{i}=u_{i}+v_{i}}
709:from corresponding operations on the
133:
1580:, Cambridge University Press, 2003.
1188:it is common to define a pointwise
1071:is a componentwise operation while
501:{\displaystyle (f+g)(x)=f(x)+g(x).}
24:
1463:
1377:
1247:, then so is the set of functions
25:
1618:
1305:Similarly, a projection operator
929:, then componentwise addition is
209:as follows: Given two functions
1578:Continuous Lattices and Domains
105:can also be defined pointwise.
1589:, which is licensed under the
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350:, and for operations of other
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1:
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1102:corresponds to the function
882:-th component of any vector
7:
1211:can be ordered by defining
737:, the set of all functions
357:
10:
1623:
1552:For order theory examples:
1169:{\displaystyle f(i)=v_{i}}
428:with the same domain and
101:of definition. Important
1607:Mathematical terminology
1517:
1434:pointwise to a function
1127:{\displaystyle f:n\to K}
785:Componentwise operations
601:{\displaystyle \lambda }
1530:Gierz et al., p. xxxiii
1203:, the set of functions
362:The pointwise addition
121:(lower plot, blue) and
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1386:
1333:pointwise relation is
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243:, define the function
201:of all functions from
130:
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61:
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1337:of functions—a
1335:pointwise convergence
1171:
1129:
1097:
1073:matrix multiplication
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924:
922:{\displaystyle v_{i}}
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809:{\displaystyle K^{n}}
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1539:Gierz, et al., p. 26
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109:Pointwise operations
95:pointwise operations
74:
60:{\displaystyle f(x)}
42:
1381:
1286:projection operator
1245:continuous lattices
1192:on functions. With
1180:Pointwise relations
862:. If we denote the
735:algebraic structure
381:{\displaystyle f+g}
138:A binary operation
18:Pointwise operation
1561:, Springer, 2005,
1505:
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1298:, where id is the
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86:{\displaystyle f.}
83:
57:
1452:
1329:An example of an
1300:identity function
1095:{\displaystyle v}
895:{\displaystyle v}
875:{\displaystyle i}
855:{\displaystyle K}
832:{\displaystyle n}
774:{\displaystyle A}
750:{\displaystyle X}
726:{\displaystyle A}
421:{\displaystyle g}
401:{\displaystyle f}
388:of two functions
134:Formal definition
16:(Redirected from
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1567:1-85233-905-5
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1557:T. S. Blyth,
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1341:of functions
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1199:
1195:
1191:
1190:partial order
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736:
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703:commutativity
700:
699:associativity
690:
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685:pointwise is
684:
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51:
45:
37:
33:
19:
1584:
1583:
1577:
1558:
1551:
1550:
1535:
1526:
1438:if for each
1328:
1321:
1316:
1310:
1309:is called a
1306:
1295:
1290:
1281:
1280:self-map on
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1238:
1234:
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1222:
1216:
1212:
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1197:
1193:
1186:order theory
1183:
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126:
94:
35:
29:
1574:D. S. Scott
1268:on a poset
759:carrier set
687:convolution
191:on the set
32:mathematics
1587:PlanetMath
1547:References
1331:infinitary
1278:idempotent
1134:such that
693:Properties
338:Commonly,
1464:∞
1461:→
1432:converges
1416:⟶
1378:∞
1119:→
839:and some
816:for some
652:⋅
649:λ
628:⋅
625:λ
596:λ
559:⋅
526:⋅
156:on a set
103:relations
36:pointwise
1601:Category
1339:sequence
1284:(i.e. a
1274:monotone
1075:is not.
733:is some
711:codomain
430:codomain
358:Examples
325:for all
69:function
67:of some
757:to the
1565:
1201:posets
713:. If
610:scalar
99:domain
1518:Notes
1392:with
1272:is a
1225:∈ A)
1080:tuple
841:field
608:is a
352:arity
181:) → (
173:) × (
1563:ISBN
1319:≤ id
1276:and
1233:) ≤
705:and
408:and
342:and
297:) =
226:and
129:=2π.
1454:lim
1442:in
1219:if
1184:In
902:as
761:of
683:not
312:),
293:))(
271:by
262:):
205:to
165:: (
119:sin
30:In
1603::
1576::
1294:≤
1259:A
1251:→
1221:(∀
1215:≤
1207:→
1196:,
1078:A
994:.
701:,
689:.
612::
354:.
330:∈
323:))
286:,
266:→
255:,
238:→
234::
221:→
217::
196:→
151:→
147:×
143::
123:ln
1593:.
1569:.
1503:.
1500:)
1497:x
1494:(
1491:f
1488:=
1485:)
1482:x
1479:(
1474:n
1470:f
1458:n
1444:X
1440:x
1436:f
1419:Y
1413:X
1410::
1405:n
1401:f
1373:1
1370:=
1367:n
1363:)
1357:n
1353:f
1349:(
1325:.
1322:A
1317:k
1307:k
1302:.
1296:c
1291:A
1282:P
1270:P
1266:c
1253:B
1249:A
1241:)
1239:x
1237:(
1235:g
1231:x
1229:(
1227:f
1223:x
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1198:B
1194:A
1162:i
1158:v
1154:=
1151:)
1148:i
1145:(
1142:f
1122:K
1116:n
1113::
1110:f
1090:v
1057:j
1054:i
1050:B
1046:+
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1038:i
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1030:=
1025:j
1022:i
1018:)
1014:B
1011:+
1008:A
1005:(
980:i
976:v
972:+
967:i
963:u
959:=
954:i
950:)
946:v
943:+
940:u
937:(
915:i
911:v
890:v
870:i
850:K
827:n
802:n
798:K
769:A
745:X
721:A
667:.
664:)
661:x
658:(
655:f
646:=
643:)
640:x
637:(
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631:f
622:(
574:.
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568:x
565:(
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547:f
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538:x
535:(
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529:g
523:f
520:(
496:.
493:)
490:x
487:(
484:g
481:+
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475:x
472:(
469:f
466:=
463:)
460:x
457:(
454:)
451:g
448:+
445:f
442:(
416:g
396:f
376:g
373:+
370:f
348:o
344:O
340:o
335:.
332:X
328:x
321:x
319:(
317:2
314:f
310:x
308:(
306:1
303:f
301:(
299:o
295:x
291:2
288:f
284:1
281:f
279:(
277:O
275:(
268:Y
264:X
260:2
257:f
253:1
250:f
248:(
246:O
240:Y
236:X
232:2
229:f
223:Y
219:X
215:1
212:f
207:Y
203:X
198:Y
194:X
189:)
187:Y
185:→
183:X
179:Y
177:→
175:X
171:Y
169:→
167:X
163:O
158:Y
153:Y
149:Y
145:Y
141:o
127:x
81:.
78:f
55:)
52:x
49:(
46:f
20:)
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