948:
783:
816:
981:
751:
894:
842:
920:
868:
715:
399:
is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the
665:
657:
1331:
Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras".
1461:
1400:
1287:
1418:
1320:
1225:
1087:
1246:
1428:
1410:
1039:
to be integrally closed and to be
Archimedean is equivalent. There is a theorem that every integrally closed
17:
1423:
1405:
925:
1060:
756:
637:
is again a partially ordered group: all operations are performed componentwise. Furthermore, every
1466:
1104:
788:
953:
1098:
1066:
370:
31:
1236:
724:
1471:
1036:
873:
821:
693:
The
Archimedean property of the real numbers can be generalized to partially ordered groups.
899:
1092:
1032:
538:
8:
847:
518:
514:
46:
1444:
1381:
700:
503:
499:
1344:
1316:
1283:
1242:
1221:
1182:
1069: – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
1371:
1340:
1308:
1275:
1213:
1174:
1048:
1031:
This property is somewhat stronger than the fact that a partially ordered group is
38:
507:
386:
1436:
590:
180:
1279:
506:. The partially ordered groups, together with this notion of morphism, form a
1455:
1217:
1186:
1072:
1044:
676:
394:
374:
54:
30:"Ordered group" redirects here. For groups with a total or linear order, see
1078:
1047:. This has to do with the fact that a directed group is embeddable into a
1040:
366:
1197:, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
545:
1300:, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
1440:
1385:
1162:
557:, where the group operation is componentwise addition, and we write (
1376:
1359:
1312:
1178:
212:
is a partially orderable group if and only if there exists a subset
638:
1063: – Group with a cyclic order respected by the group operation
358:. Being unperforated means there is no "gap" in the positive cone
1266:, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
553:
531:
377:, i.e. any two elements have a least upper bound, then it is a
1051:
lattice-ordered group if and only if it is integrally closed.
170:. So we can reduce the partial order to a monadic property:
645:
is a partially ordered group: it inherits the order from
204:, the existence of a positive cone specifies an order on
1255:
V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish),
513:
Partially ordered groups are used in the definition of
956:
928:
902:
876:
850:
824:
791:
759:
727:
703:
61:; in other words, "≤" has the property that, for all
1083:
Pages displaying wikidata descriptions as a fallback
1075: – Algebraic object with an ordered structure
975:
942:
914:
888:
862:
836:
810:
777:
745:
709:
551:A typical example of a partially ordered group is
1364:Transactions of the American Mathematical Society
1453:
1445:Creative Commons Attribution/Share-Alike License
629:is some set, then the set of all functions from
1269:
1259:, Halsted Press (John Wiley & Sons), 1974.
486:are two partially ordered groups, a map from
1101: – Ring with a compatible partial order
1357:
1107: – Partially ordered topological space
664:is a stably finite unital C*-algebra, then
658:approximately finite-dimensional C*-algebra
1081: – ring with a compatible total order
1375:
1305:Lattices and Ordered Algebraic Structures
1270:Kopytov, V. M.; Medvedev, N. Ya. (1994).
1095: – Vector space with a partial order
936:
610:(in the usual order of integers) for all
1435:This article incorporates material from
1160:
1138:
1416:
1398:
1330:
1155:Lattice Ordered Groups: an Introduction
1123:
1121:
14:
1454:
1234:
1207:
1127:
27:Group with a compatible partial order
1272:The Theory of Lattice-Ordered Groups
1195:The Theory of Lattice-Ordered Groups
1118:
987:
697:Property: A partially ordered group
496:morphism of partially ordered groups
1262:V. M. Kopytov and N. Ya. Medvedev,
1202:Partially Ordered Algebraic Systems
154:By translation invariance, we have
24:
1352:
25:
1483:
1392:
1358:Everett, C. J.; Ulam, S. (1945).
943:{\displaystyle n\in \mathbb {Z} }
625:is a partially ordered group and
373:. If the order on the group is a
1088:Ordered topological vector space
385:, though usually typeset with a
365:If the order on the group is a
1443:, which is licensed under the
1132:
688:
13:
1:
1296:R. B. Mura and A. Rhemtulla,
1147:
778:{\displaystyle e\leq a\leq b}
683:
1462:Ordered algebraic structures
1345:10.1016/0021-8693(76)90242-8
541:is a partially ordered group
401:Riesz interpolation property
7:
1424:Encyclopedia of Mathematics
1406:Encyclopedia of Mathematics
1054:
811:{\displaystyle a^{n}\leq b}
524:
10:
1488:
1210:Ordered Permutation Groups
1161:Birkhoff, Garrett (1942).
992:A partially ordered group
976:{\displaystyle b<a^{n}}
548:is a lattice-ordered group
369:, then it is said to be a
346:for some positive integer
322:A partially ordered group
136:. The set of elements 0 ≤
29:
1401:"Partially ordered group"
1280:10.1007/978-94-015-8304-6
1167:The Annals of Mathematics
1153:M. Anderson and T. Feil,
675:) is a partially ordered
1238:Partially Ordered Groups
1235:Glass, A. M. W. (1999).
1218:10.1017/CBO9780511721243
1208:Glass, A. M. W. (1982).
1163:"Lattice-Ordered Groups"
1061:Cyclically ordered group
746:{\displaystyle a,b\in G}
660:, or more generally, if
1437:partially ordered group
1419:"Lattice-ordered group"
1417:Kopytov, V.M. (2001) ,
1399:Kopytov, V.M. (2001) ,
1204:, Pergamon Press, 1963.
1111:
1105:Partially ordered space
889:{\displaystyle a\neq e}
837:{\displaystyle n\geq 1}
43:partially ordered group
1307:. Universitext. 2005.
1099:Partially ordered ring
1067:Linearly ordered group
977:
944:
916:
915:{\displaystyle b\in G}
890:
864:
838:
812:
779:
747:
711:
534:with their usual order
371:linearly ordered group
200:For the general group
140:is often denoted with
32:Linearly ordered group
1037:lattice-ordered group
978:
945:
917:
891:
870:. Equivalently, when
865:
839:
813:
780:
748:
712:
379:lattice-ordered group
59:translation-invariant
53:, +) equipped with a
1264:Right-ordered groups
1257:Fully Ordered Groups
1093:Ordered vector space
1000:if for all elements
954:
926:
900:
874:
848:
822:
789:
757:
725:
701:
539:ordered vector space
449:, then there exists
162:if and only if 0 ≤ -
144:, and is called the
1360:"On Ordered Groups"
863:{\displaystyle a=e}
621:More generally, if
326:with positive cone
1333:Journal of Algebra
1157:, D. Reidel, 1988.
973:
940:
912:
886:
860:
834:
808:
775:
743:
707:
504:monotonic function
500:group homomorphism
1289:978-90-481-4474-7
1043:group is already
998:integrally closed
988:Integrally closed
710:{\displaystyle G}
679:. (Elliott, 1976)
146:positive cone of
16:(Redirected from
1479:
1431:
1413:
1389:
1379:
1348:
1326:
1298:Orderable groups
1293:
1252:
1231:
1190:
1141:
1136:
1130:
1125:
1084:
1020:for all natural
982:
980:
979:
974:
972:
971:
949:
947:
946:
941:
939:
922:, there is some
921:
919:
918:
913:
895:
893:
892:
887:
869:
867:
866:
861:
843:
841:
840:
835:
817:
815:
814:
809:
801:
800:
784:
782:
781:
776:
752:
750:
749:
744:
716:
714:
713:
708:
498:if it is both a
431:are elements of
197:
179:
39:abstract algebra
21:
1487:
1486:
1482:
1481:
1480:
1478:
1477:
1476:
1452:
1451:
1395:
1377:10.2307/1990202
1355:
1353:Further reading
1323:
1313:10.1007/b139095
1303:
1290:
1249:
1228:
1179:10.2307/1968871
1150:
1145:
1144:
1139:Birkhoff (1942)
1137:
1133:
1126:
1119:
1114:
1082:
1057:
1035:, though for a
990:
967:
963:
955:
952:
951:
935:
927:
924:
923:
901:
898:
897:
896:, then for any
875:
872:
871:
849:
846:
845:
823:
820:
819:
796:
792:
790:
787:
786:
758:
755:
754:
726:
723:
722:
702:
699:
698:
691:
686:
669:
609:
600:
588:
579:
572:
563:
527:
473:
462:
447:
440:
430:
423:
416:
409:
183:
171:
35:
28:
23:
22:
15:
12:
11:
5:
1485:
1475:
1474:
1469:
1467:Ordered groups
1464:
1450:
1449:
1432:
1414:
1394:
1393:External links
1391:
1370:(2): 208–216.
1354:
1351:
1350:
1349:
1328:
1321:
1301:
1294:
1288:
1267:
1260:
1253:
1247:
1232:
1226:
1205:
1198:
1193:M. R. Darnel,
1191:
1158:
1149:
1146:
1143:
1142:
1131:
1116:
1115:
1113:
1110:
1109:
1108:
1102:
1096:
1090:
1085:
1076:
1070:
1064:
1056:
1053:
989:
986:
985:
984:
970:
966:
962:
959:
938:
934:
931:
911:
908:
905:
885:
882:
879:
859:
856:
853:
833:
830:
827:
807:
804:
799:
795:
774:
771:
768:
765:
762:
742:
739:
736:
733:
730:
706:
690:
687:
685:
682:
681:
680:
667:
650:
619:
605:
596:
591:if and only if
584:
577:
568:
561:
549:
542:
535:
526:
523:
471:
460:
445:
438:
428:
421:
414:
407:
330:is said to be
320:
319:
296:
262:
232:
181:if and only if
26:
9:
6:
4:
3:
2:
1484:
1473:
1470:
1468:
1465:
1463:
1460:
1459:
1457:
1448:
1446:
1442:
1438:
1433:
1430:
1426:
1425:
1420:
1415:
1412:
1408:
1407:
1402:
1397:
1396:
1390:
1387:
1383:
1378:
1373:
1369:
1365:
1361:
1346:
1342:
1338:
1334:
1329:
1324:
1322:1-85233-905-5
1318:
1314:
1310:
1306:
1302:
1299:
1295:
1291:
1285:
1281:
1277:
1273:
1268:
1265:
1261:
1258:
1254:
1250:
1244:
1240:
1239:
1233:
1229:
1227:9780521241908
1223:
1219:
1215:
1211:
1206:
1203:
1199:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1159:
1156:
1152:
1151:
1140:
1135:
1129:
1124:
1122:
1117:
1106:
1103:
1100:
1097:
1094:
1091:
1089:
1086:
1080:
1077:
1074:
1073:Ordered field
1071:
1068:
1065:
1062:
1059:
1058:
1052:
1050:
1046:
1042:
1038:
1034:
1029:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
968:
964:
960:
957:
932:
929:
909:
906:
903:
883:
880:
877:
857:
854:
851:
831:
828:
825:
805:
802:
797:
793:
772:
769:
766:
763:
760:
740:
737:
734:
731:
728:
721:when for any
720:
704:
696:
695:
694:
678:
677:abelian group
674:
670:
663:
659:
655:
651:
648:
644:
640:
636:
632:
628:
624:
620:
617:
613:
608:
604:
599:
595:
592:
587:
583:
576:
571:
567:
560:
556:
555:
550:
547:
543:
540:
536:
533:
529:
528:
522:
520:
516:
511:
509:
505:
501:
497:
493:
489:
485:
481:
476:
474:
467:
463:
456:
452:
448:
441:
434:
427:
420:
413:
406:
402:
398:
396:
390:
389:l: ℓ-group).
388:
384:
380:
376:
375:lattice order
372:
368:
363:
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
317:
313:
309:
305:
301:
297:
295:
291:
287:
283:
279:
275:
271:
267:
263:
261:
257:
253:
249:
245:
241:
237:
233:
231:
227:
226:
225:
223:
219:
215:
211:
207:
203:
198:
195:
191:
187:
182:
178:
174:
169:
165:
161:
157:
152:
150:
149:
143:
139:
135:
131:
127:
123:
118:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
55:partial order
52:
48:
44:
40:
33:
19:
18:Ordered group
1472:Order theory
1434:
1422:
1404:
1367:
1363:
1356:
1336:
1332:
1304:
1297:
1271:
1263:
1256:
1237:
1209:
1201:
1194:
1170:
1166:
1154:
1134:
1128:Glass (1999)
1079:Ordered ring
1030:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
991:
718:
692:
672:
661:
653:
646:
642:
634:
630:
626:
622:
615:
611:
606:
602:
597:
593:
585:
581:
574:
569:
565:
558:
552:
512:
495:
491:
487:
483:
479:
477:
469:
465:
458:
454:
450:
443:
436:
432:
425:
418:
411:
404:
400:
393:
391:
382:
378:
367:linear order
364:
359:
355:
351:
347:
343:
339:
335:
332:unperforated
331:
327:
323:
321:
315:
311:
307:
303:
299:
293:
289:
285:
281:
277:
273:
269:
265:
259:
255:
251:
247:
243:
239:
235:
229:
221:
217:
213:
209:
205:
201:
199:
193:
189:
185:
176:
172:
167:
163:
159:
155:
153:
147:
145:
141:
137:
133:
129:
125:
121:
119:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
57:"≤" that is
50:
42:
36:
1033:Archimedean
719:Archimedean
689:Archimedean
546:Riesz space
224:such that:
120:An element
1456:Categories
1441:PlanetMath
1327:, chap. 9.
1248:981449609X
1200:L. Fuchs,
1173:(2): 313.
1148:References
996:is called
950:such that
717:is called
684:Properties
515:valuations
457:such that
216:(which is
208:. A group
128:is called
1429:EMS Press
1411:EMS Press
1339:: 29–44.
1187:0003-486X
933:∈
907:∈
881:≠
829:≥
803:≤
770:≤
764:≤
738:∈
614:= 1,...,
381:(shortly
288:for each
1055:See also
1049:complete
1041:directed
818:for all
639:subgroup
532:integers
525:Examples
508:category
350:implies
130:positive
1386:1990202
1045:abelian
383:l-group
132:if 0 ≤
1384:
1319:
1286:
1245:
1224:
1185:
656:is an
519:fields
502:and a
387:script
272:then -
69:, and
1382:JSTOR
1028:≤ 1.
1024:then
1012:, if
844:then
753:, if
580:,...,
573:) ≤ (
564:,...,
494:is a
403:: if
397:group
395:Riesz
314:then
306:and -
250:then
220:) of
85:then
77:, if
47:group
45:is a
1317:ISBN
1284:ISBN
1243:ISBN
1222:ISBN
1183:ISSN
1112:Note
1004:and
961:<
785:and
530:The
482:and
435:and
242:and
228:0 ∈
101:and
41:, a
1439:on
1372:doi
1341:doi
1309:doi
1276:doi
1214:doi
1175:doi
1008:of
652:If
641:of
633:to
537:An
517:of
490:to
478:If
334:if
318:= 0
298:if
292:of
264:if
234:if
124:of
73:in
37:In
1458::
1427:,
1421:,
1409:,
1403:,
1380:.
1368:57
1366:.
1362:.
1337:38
1335:.
1315:.
1282:.
1274:.
1241:.
1220:.
1212:.
1181:.
1171:43
1169:.
1165:.
1120:^
1016:≤
601:≤
589:)
544:A
521:.
510:.
475:.
468:≤
464:≤
453:∈
442:≤
424:,
417:,
410:,
392:A
362:.
354:∈
342:∈
338:·
310:∈
302:∈
284:∈
280:+
276:+
268:∈
258:∈
254:+
246:∈
238:∈
192:∈
188:+
175:≤
166:+
158:≤
151:.
117:.
109:≤
97:+
93:≤
89:+
81:≤
65:,
1447:.
1388:.
1374::
1347:.
1343::
1325:.
1311::
1292:.
1278::
1251:.
1230:.
1216::
1189:.
1177::
1026:a
1022:n
1018:b
1014:a
1010:G
1006:b
1002:a
994:G
983:.
969:n
965:a
958:b
937:Z
930:n
910:G
904:b
884:e
878:a
858:e
855:=
852:a
832:1
826:n
806:b
798:n
794:a
773:b
767:a
761:e
741:G
735:b
732:,
729:a
705:G
673:A
671:(
668:0
666:K
662:A
654:A
649:.
647:G
643:G
635:G
631:X
627:X
623:G
618:.
616:n
612:i
607:i
603:b
598:i
594:a
586:n
582:b
578:1
575:b
570:n
566:a
562:1
559:a
554:Z
492:H
488:G
484:H
480:G
472:j
470:y
466:z
461:i
459:x
455:G
451:z
446:j
444:y
439:i
437:x
433:G
429:2
426:y
422:1
419:y
415:2
412:x
408:1
405:x
360:G
356:G
352:g
348:n
344:G
340:g
336:n
328:G
324:G
316:a
312:H
308:a
304:H
300:a
294:G
290:x
286:H
282:x
278:a
274:x
270:H
266:a
260:H
256:b
252:a
248:H
244:b
240:H
236:a
230:H
222:G
218:G
214:H
210:G
206:G
202:G
196:.
194:G
190:b
186:a
184:-
177:b
173:a
168:b
164:a
160:b
156:a
148:G
142:G
138:x
134:x
126:G
122:x
115:b
113:+
111:g
107:a
105:+
103:g
99:g
95:b
91:g
87:a
83:b
79:a
75:G
71:g
67:b
63:a
51:G
49:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.