334:
Every history preserving function and every future preserving function is also order preserving, but not vice versa. In the theory of dynamical systems, history preserving maps capture the intuition that the behavior in one system is a
1214:
1197:
32:
by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering
727:
563:
532:
1044:
406:
the
Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an
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we find that this set is prefix ordered by the subset relation ⊆, and furthermore, that the function
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Ferlez, James; Cleaveland, Rance; Marcus, Steve (2014). "Generalized
Synchronization Trees".
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1126:
871:
834:
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804:
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Proceedings of the 9th
International Workshop on Developments in Computational Models (DCM)
339:
of the behavior in another. Furthermore, functions that are history and future preserving
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191:, the most important type of functions between prefix orders are so-called
21:
1202:
895:
774:
639:
45:
17:
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477:
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517:. Lecture Notes in Computer Science. Vol. 8412. pp. 304–319.
1221:
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of the original prefix orders. The union of two prefix orders is the
60:
1016:
883:
634:
347:
between systems, and thus the intuition that a given refinement is
48:. In this case, the elements of the set are usually referred to as
494:
59:
stems from the prefix order on words, which is a special kind of
541:
515:
Foundations of
Software Science and Computation Structures
478:"The Categorical Limit of a Sequence of Dynamical Systems"
63:
relation and, because of its discrete character, a tree.
512:
402:
of prefix orders leads to a notion of product that is
446:
the maximum element in terms of the order on P (i.e.
187:
While between partial orders it is usual to consider
182:
426:. Furthermore, if for a given prefix ordered set
1239:
422:Any bijective history preserving function is an
462:"Prefix Orders as a General Model of Dynamics"
557:
358:of a history preserving function is always a
211:is the (by definition totally ordered) set
1215:Positive cone of a partially ordered group
564:
550:
522:
503:
493:
1198:Positive cone of an ordered vector space
475:
459:
482:EPTCS 120: Proceedings EXPRESS/SOS 2013
28:generalizes the intuitive concept of a
1240:
195:functions. Given a prefix ordered set
545:
389:
66:
13:
725:Properties & Types (
394:Taking history preserving maps as
14:
1269:
1181:Positive cone of an ordered field
351:with respect to a specification.
1035:Ordered topological vector space
571:
414:, as it is with partial orders.
307:is future preserving if for all
183:Functions between prefix orders
417:
1:
992:Series-parallel partial order
453:
671:Cantor's isomorphism theorem
524:10.1007/978-3-642-54830-7_20
287:is the (prefix ordered) set
7:
711:Szpilrajn extension theorem
686:Hausdorff maximal principle
661:Boolean prime ideal theorem
36:as a set of functions from
10:
1274:
1057:Topological vector lattice
189:order-preserving functions
1087:
1015:
954:
724:
653:
602:
579:
476:Cuijpers, Pieter (2013).
460:Cuijpers, Pieter (2013).
438:is an isomorphism, where
251:if and only if for every
666:Cantor–Bernstein theorem
1258:Trees (data structures)
1210:Partially ordered group
1030:Specialization preorder
362:subset, where a subset
275:)−. Similarly, a
696:Kruskal's tree theorem
691:Knaster–Tarski theorem
681:Dushnik–Miller theorem
408:arbitrary interleaving
343:capture the notion of
239:between prefix orders
442:returns for each set
430:we construct the set
1188:Ordered vector space
178:(downward totality).
1026:Alexandrov topology
972:Lexicographic order
931:Well-quasi-ordering
505:10.4204/EPTCS.120.7
42:totally-ordered set
1007:Transitive closure
967:Converse/Transpose
676:Dilworth's theorem
249:history preserving
193:history preserving
26:prefix ordered set
1248:Dynamical systems
1235:
1234:
1193:Partially ordered
1002:Symmetric closure
987:Reflexive closure
730:
534:978-3-642-54829-1
471:. pp. 25–29.
432:P- ≜ { p- | p∈ P}
424:order isomorphism
390:Product and union
67:Formal definition
34:dynamical systems
1265:
977:Linear extension
726:
706:Mirsky's theorem
566:
559:
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118:, we have that:
102:, i.e., for all
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1227:Young's lattice
1083:
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950:
800:Heyting algebra
748:Boolean algebra
720:
701:Laver's theorem
649:
615:Boolean algebra
610:Binary relation
598:
575:
570:
535:
464:
456:
420:
392:
185:
159:(transitivity);
144:(antisymmetry);
77:binary relation
69:
52:of the system.
12:
11:
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1154:Order morphism
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1072:Locally convex
1069:
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1052:Order topology
1049:
1048:
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1045:Order topology
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857:Chain-complete
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510:
473:
455:
452:
419:
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412:disjoint union
391:
388:
227:}. A function
184:
181:
180:
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145:
126:
125:(reflexivity);
100:downward total
68:
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9:
6:
4:
3:
2:
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1208:
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1201:
1199:
1196:
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1179:
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1177:
1176:Ordered field
1174:
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1132:Hasse diagram
1130:
1128:
1125:
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1116:
1113:
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1111:
1110:Comparability
1108:
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995:
993:
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988:
985:
983:
982:Product order
980:
978:
975:
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968:
965:
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957:
955:Constructions
953:
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868:
865:
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852:Partial order
850:
848:
845:
841:
840:Join and meet
838:
836:
833:
831:
828:
826:
823:
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811:
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766:
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754:
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750:
749:
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741:
739:
738:Antisymmetric
736:
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729:
723:
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662:
659:
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645:Weak ordering
643:
641:
638:
636:
633:
631:
630:Partial order
628:
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623:
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548:
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525:
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501:
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487:
483:
479:
474:
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463:
458:
457:
451:
449:
445:
441:
437:
433:
429:
425:
415:
413:
409:
405:
401:
397:
387:
385:
381:
377:
373:
369:
368:prefix closed
365:
361:
360:prefix closed
357:
352:
350:
346:
342:
338:
332:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
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177:
173:
169:
165:
161:
158:
154:
150:
146:
143:
139:
135:
131:
127:
124:
121:
120:
119:
117:
113:
109:
105:
101:
97:
93:
89:
88:antisymmetric
85:
82:
78:
74:
64:
62:
58:
53:
51:
47:
43:
39:
35:
31:
27:
23:
20:, especially
19:
1253:Order theory
1019:& Orders
997:Star product
926:Well-founded
879:Prefix order
878:
835:Distributive
825:Complemented
795:Foundational
760:Completeness
716:Zorn's lemma
620:Cyclic order
603:Key concepts
573:Order theory
514:
485:
481:
468:
447:
443:
439:
435:
431:
427:
421:
407:
403:
395:
393:
383:
379:
375:
371:
367:
363:
353:
348:
345:bisimulation
336:
333:
328:
324:
320:
316:
312:
308:
304:
300:
296:
292:
288:
284:
280:
276:
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268:
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256:
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236:
232:
228:
224:
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208:
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200:
196:
192:
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175:
171:
167:
163:
156:
152:
148:
141:
137:
133:
129:
122:
115:
111:
107:
103:
99:
83:
79:"≤" over a
73:prefix order
72:
70:
57:prefix order
56:
54:
49:
37:
25:
22:order theory
15:
1203:Riesz space
1164:Isomorphism
1040:Normal cone
962:Composition
896:Semilattice
805:Homogeneous
790:Equivalence
640:Total order
448:max(p-) ≜ p
436:max: P- → P
418:Isomorphism
370:if for all
341:surjections
279:of a point
267:−) =
215:− = {
203:of a point
46:phase space
18:mathematics
1242:Categories
1171:Order type
1105:Cofinality
946:Well-order
921:Transitive
810:Idempotent
743:Asymmetric
454:References
337:refinement
92:transitive
50:executions
44:) to some
1222:Upper set
1159:Embedding
1095:Antichain
916:Tolerance
906:Symmetric
901:Semiorder
847:Reflexive
765:Connected
495:1307.7445
488:: 78–92.
396:morphisms
96:reflexive
86:which is
61:substring
55:The name
1017:Topology
884:Preorder
867:Eulerian
830:Complete
780:Directed
770:Covering
635:Preorder
594:Category
589:Glossary
400:category
382:we find
315:we find
259:we find
247:is then
1122:Duality
1100:Cofinal
1088:Related
1067:Fréchet
944:)
820:Bounded
815:Lattice
788:)
786:Partial
654:Results
625:Lattice
398:in the
372:s,t ∈ P
349:correct
201:history
1147:Subnet
1127:Filter
1077:Normed
1062:Banach
1028:&
935:Better
872:Strict
862:Graded
753:topics
584:Topics
531:
440:max(S)
303:} and
277:future
110:, and
98:, and
1137:Ideal
1115:Graph
911:Total
889:Total
775:Dense
490:arXiv
465:(PDF)
374:with
364:S ⊆ P
356:range
323:+) =
291:+ = {
176:b ≤ a
172:a ≤ b
170:then
168:b ≤ c
164:a ≤ c
157:a ≤ c
155:then
153:b ≤ c
149:a ≤ b
136:then
134:b ≤ a
130:a ≤ b
123:a ≤ a
75:is a
728:list
529:ISBN
444:S∈P-
378:and
354:The
331:)+.
243:and
199:, a
166:and
151:and
132:and
38:time
30:tree
24:, a
1142:Net
942:Pre
519:doi
500:doi
486:120
450:).
404:not
384:s∈S
380:s≤t
376:t∈S
366:is
174:or
162:if
147:if
128:if
114:in
81:set
40:(a
16:In
1244::
527:.
498:.
484:.
480:.
467:.
386:.
299:≤
295:|
235:→
231::
223:≤
219:|
140:=
106:,
94:,
90:,
71:A
940:(
937:)
933:(
784:(
731:)
565:e
558:t
551:v
537:.
521::
508:.
502::
492::
428:P
329:p
327:(
325:f
321:p
319:(
317:f
313:P
311:∈
309:p
305:f
301:q
297:p
293:q
289:p
285:P
283:∈
281:p
273:p
271:(
269:f
265:p
263:(
261:f
257:P
255:∈
253:p
245:Q
241:P
237:Q
233:P
229:f
225:p
221:q
217:q
213:p
209:P
207:∈
205:p
197:P
142:b
138:a
116:P
112:c
108:b
104:a
84:P
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