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