1386:, and a basis is called homogeneous if any two elements are perspective. The order of a lattice need not be unique; for example, any lattice has order 1. The condition that the lattice has order at least 4 corresponds to the condition that the dimension is at least 3 in the Veblen–Young theorem, as a projective space has dimension at least 3 if and only if it has a set of at least 4 independent points.
259:
656:
432:
1097:
126:
Menger and
Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms.
166:
76:
536:
114:
with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the
1256:
to the projective geometry of a vector space over a division ring. This can be restated as saying that the subspaces in the projective geometry correspond to the
358:
Finite-dimensional complex projective space, or rather its set of linear subspaces, is a continuous geometry, with dimensions taking values in the discrete set
696:
466:
108:
551:
1111:
if and only if they are perspective, so it gives an injection from the equivalence classes to a subset of the unit interval. The dimension function
1627:
361:
436:
The projections of a finite type II von
Neumann algebra form a continuous geometry with dimensions taking values in the unit interval
1032:
995:
contains a minimal nonzero element, or an infinite sequence of nonzero elements each of which is at most half the preceding one.
1842:
1784:
1448:
718:, Part I). These results are similar to, and were motivated by, von Neumann's work on projections in von Neumann algebras.
1263:
Neumann generalized this to continuous geometries, and more generally to complemented modular lattices, as follows (
254:{\displaystyle {\Big (}\bigwedge _{\alpha \in A}a_{\alpha }{\Big )}\lor b=\bigwedge _{\alpha }(a_{\alpha }\lor b)}
1488:
39:
1899:
1476:
1440:
1894:
1466:
1389:
Conversely, the principal right ideals of a von
Neumann regular ring form a complemented modular lattice (
1471:
1439:, American Mathematical Society Colloquium Publications, vol. 25 (3rd ed.), Providence, R.I.:
1776:
115:
508:
1276:
1249:
666:
between 0 and 1. Its completion is a continuous geometry containing elements of every dimension in
1904:
1578:
1878:
1860:
1815:
1794:
1711:
1636:
1615:
1558:
1519:
1458:
1257:
748:
475:
1733:
1692:
8:
486:
111:
21:
1715:
1640:
1529:(1985), "Books in Review: A survey of John von Neumann's books on continuous geometry",
1750:
1737:
1675:
1662:
1603:
1562:
669:
439:
348:
is irreducible: this means that the only elements with unique complements are 0 and 1.
81:
1848:
1838:
1780:
1755:
1680:
1654:
1595:
1566:
1546:
1507:
1444:
1822:
1801:
1766:
1745:
1729:
1719:
1699:
1688:
1670:
1644:
1622:
1587:
1576:(1955), "Any orthocomplemented complete modular lattice is a continuous geometry",
1573:
1538:
1497:
1430:
155:
25:
1874:
1868:
1856:
1826:
1811:
1805:
1790:
1770:
1611:
1554:
1526:
1515:
1483:
1454:
1434:
663:
145:
131:
1628:
Proceedings of the
National Academy of Sciences of the United States of America
651:{\displaystyle PG(F)\subset PG(F^{2})\subset PG(F^{4})\subset PG(F^{8})\cdots }
1888:
1852:
1658:
1599:
1550:
1511:
490:
1759:
1684:
1502:
539:
266:
1724:
1649:
1486:(1960), "Introduction to von Neumann algebras and continuous geometry",
36:), where instead of the dimension of a subspace being in a discrete set
1607:
1542:
1267:, Part II). His theorem states that if a complemented modular lattice
308:
has a complement (not necessarily unique). A complement of an element
1741:
1666:
1417:
1591:
1253:
161:
The lattice operations ∧, ∨ satisfy a certain continuity property,
427:{\displaystyle \{0,1/{\textit {n}}\,,2/{\textit {n}}\,,\dots ,1\}}
1807:
Collected works. Vol. IV: Continuous geometry and other topics
1092:{\displaystyle 0,1/{\textit {n}}\,,2/{\textit {n}}\,,\dots ,1}
1252:
states that a projective geometry of dimension at least 3 is
925:
not {0} the integer is defined to be the unique integer
1283:
then the von
Neumann regular ring can be taken to be an
991:
runs through a minimal sequence: this means that either
338:, where 0 and 1 are the minimal and maximal elements of
1408:
is a complemented modular lattice. Neumann showed that
1029:
can be the whole unit interval, or the set of numbers
1827:"Continuous geometries with a transition probability"
1035:
672:
554:
511:
442:
364:
169:
84:
42:
1091:
690:
650:
538:that multiplies dimensions by 2. So we can take a
530:
497:, then there is a natural map from the lattice PG(
478:complete modular lattice is a continuous geometry.
460:
426:
253:
102:
70:
1017:} and {1} are the equivalence classes containing
755:; the proof that it is transitive is quite hard.
205:
172:
1886:
1308:. Here a complemented modular lattice has order
747:, if they have a common complement. This is an
662:This has a dimension function taking values all
110:. Von Neumann was motivated by his discovery of
1870:Complemented modular lattices and regular rings
1404:its lattice of principal right ideals, so that
714:This section summarizes some of the results of
299:, and the same condition with ∧ and ∨ reversed.
1702:(1936b), "Examples of continuous geometries",
1275:correspond to the principal right ideals of a
833:to the unit interval is defined as follows.
702:, and is called the continuous geometry over
1831:Memoirs of the American Mathematical Society
1698:
699:
421:
365:
78:, it can be an element of the unit interval
1821:
1800:
1765:
1621:
1390:
1271:has order at least 4, then the elements of
1264:
1243:
715:
33:
29:
1866:
1279:. More precisely if the lattice has order
1260:of a matrix algebra over a division ring.
874:is defined to be the equivalence class of
1749:
1723:
1674:
1648:
1572:
1501:
1076:
1057:
471:
408:
389:
1525:
1482:
1464:
1429:
1412:is a continuous geometry if and only if
1304:) over another von Neumann regular ring
71:{\displaystyle 0,1,\dots ,{\textit {n}}}
894:is not defined. For a positive integer
1887:
1775:, Princeton Landmarks in Mathematics,
770:have a total order on them defined by
1071:
1052:
403:
384:
63:
698:. This geometry was constructed by
13:
1400:is a von Neumann regular ring and
14:
1916:
1825:(1981) , Halperin, Israel (ed.),
1312:if it has a homogeneous basis of
1625:(1936), "Continuous geometry",
505:to the lattice of subspaces of
1489:Canadian Mathematical Bulletin
983:is defined to be the limit of
806:. (This need not hold for all
685:
673:
642:
629:
617:
604:
592:
579:
567:
561:
531:{\displaystyle V\otimes F^{2}}
455:
443:
248:
229:
137:with the following properties
97:
85:
1:
1873:, London: Oliver & Boyd,
1441:American Mathematical Society
1423:
121:
1248:In projective geometry, the
902:is defined to be the sum of
709:
7:
1804:(1962), Taub, A. H. (ed.),
1472:Encyclopedia of Mathematics
1416:is an irreducible complete
1316:elements, where a basis is
352:
130:A continuous geometry is a
10:
1921:
1867:Skornyakov, L. A. (1964),
1810:, Oxford: Pergamon Press,
1777:Princeton University Press
1704:Proc. Natl. Acad. Sci. USA
1107:have the same image under
1099:for some positive integer
116:hyperfinite type II factor
20:is an analogue of complex
910:, if this sum is defined.
485:is a vector space over a
1465:Fofanova, T.S. (2001) ,
1393:, Part II theorem 2.4).
1277:von Neumann regular ring
1244:Coordinatization theorem
971:not {0} the real number
959:For equivalence classes
913:For equivalence classes
758:The equivalence classes
837:If equivalence classes
825:The dimension function
1503:10.4153/CMB-1960-034-5
1467:"Orthomodular lattice"
1258:principal right ideals
1093:
692:
652:
532:
462:
428:
255:
104:
72:
1725:10.1073/pnas.22.2.101
1579:Annals of Mathematics
1094:
693:
653:
533:
463:
429:
256:
105:
73:
1900:Von Neumann algebras
1650:10.1073/pnas.22.2.92
1250:Veblen–Young theorem
1115:has the properties:
1033:
884:. Otherwise the sum
749:equivalence relation
670:
552:
509:
440:
362:
167:
112:von Neumann algebras
82:
40:
1895:Projective geometry
1772:Continuous geometry
1716:1936PNAS...22..101N
1641:1936PNAS...22...92N
1005:) is defined to be
700:von Neumann (1936b)
22:projective geometry
18:continuous geometry
1543:10.1007/BF00383607
1103:. Two elements of
1089:
688:
648:
528:
501:) of subspaces of
458:
424:
251:
228:
192:
100:
68:
1844:978-0-8218-2252-4
1823:von Neumann, John
1802:von Neumann, John
1786:978-0-691-05893-1
1767:von Neumann, John
1700:von Neumann, John
1623:von Neumann, John
1582:, Second Series,
1574:Kaplansky, Irving
1450:978-0-8218-1025-5
1431:Birkhoff, Garrett
1073:
1054:
845:contain elements
780:if there is some
716:von Neumann (1998
476:orthocomplemented
405:
386:
304:Every element in
219:
177:
65:
1912:
1881:
1863:
1818:
1797:
1762:
1753:
1727:
1695:
1678:
1652:
1618:
1569:
1527:Halperin, Israel
1522:
1505:
1484:Halperin, Israel
1479:
1461:
1391:von Neumann 1998
1385:
1366:
1356:
1265:von Neumann 1998
1239:
1226:
1219:
1208:
1201:
1147:
1128:
1098:
1096:
1095:
1090:
1075:
1074:
1068:
1056:
1055:
1049:
1012:
986:
982:
955:
945:
931:
893:
883:
873:
863:
805:
779:
746:
697:
695:
694:
691:{\displaystyle }
689:
664:dyadic rationals
657:
655:
654:
649:
641:
640:
616:
615:
591:
590:
537:
535:
534:
529:
527:
526:
474:showed that any
472:Kaplansky (1955)
467:
465:
464:
461:{\displaystyle }
459:
433:
431:
430:
425:
407:
406:
400:
388:
387:
381:
337:
326:
298:
278:
260:
258:
257:
252:
241:
240:
227:
209:
208:
202:
201:
191:
176:
175:
109:
107:
106:
103:{\displaystyle }
101:
77:
75:
74:
69:
67:
66:
16:In mathematics,
1920:
1919:
1915:
1914:
1913:
1911:
1910:
1909:
1885:
1884:
1845:
1787:
1592:10.2307/1969811
1451:
1426:
1383:
1374:
1368:
1358:
1354:
1345:
1337:
1335:
1326:
1299:
1246:
1229:
1221:
1220:if and only if
1210:
1203:
1202:if and only if
1192:
1130:
1120:
1070:
1069:
1064:
1051:
1050:
1045:
1034:
1031:
1030:
1006:
984:
972:
947:
933:
926:
885:
875:
865:
864:then their sum
854:
797:
771:
738:
712:
671:
668:
667:
636:
632:
611:
607:
586:
582:
553:
550:
549:
522:
518:
510:
507:
506:
441:
438:
437:
402:
401:
396:
383:
382:
377:
363:
360:
359:
355:
328:
317:
297:
288:
280:
270:
236:
232:
223:
204:
203:
197:
193:
181:
171:
170:
168:
165:
164:
124:
83:
80:
79:
62:
61:
41:
38:
37:
24:introduced by
12:
11:
5:
1918:
1908:
1907:
1905:Lattice theory
1902:
1897:
1883:
1882:
1864:
1843:
1819:
1798:
1785:
1763:
1710:(2): 101–108,
1696:
1619:
1586:(3): 524–541,
1570:
1537:(3): 301–305,
1523:
1496:(3): 273–288,
1480:
1462:
1449:
1436:Lattice theory
1425:
1422:
1396:Suppose that
1379:
1372:
1350:
1341:
1331:
1324:
1295:
1245:
1242:
1241:
1240:
1227:
1190:
1148:
1088:
1085:
1082:
1079:
1067:
1063:
1060:
1048:
1044:
1041:
1038:
1023:
1022:
996:
957:
911:
898:, the product
711:
708:
707:
706:
687:
684:
681:
678:
675:
660:
659:
658:
647:
644:
639:
635:
631:
628:
625:
622:
619:
614:
610:
606:
603:
600:
597:
594:
589:
585:
581:
578:
575:
572:
569:
566:
563:
560:
557:
544:
543:
525:
521:
517:
514:
479:
469:
457:
454:
451:
448:
445:
434:
423:
420:
417:
414:
411:
399:
395:
392:
380:
376:
373:
370:
367:
354:
351:
350:
349:
343:
312:is an element
302:
301:
300:
293:
284:
250:
247:
244:
239:
235:
231:
226:
222:
218:
215:
212:
207:
200:
196:
190:
187:
184:
180:
174:
159:
149:
123:
120:
99:
96:
93:
90:
87:
60:
57:
54:
51:
48:
45:
9:
6:
4:
3:
2:
1917:
1906:
1903:
1901:
1898:
1896:
1893:
1892:
1890:
1880:
1876:
1872:
1871:
1865:
1862:
1858:
1854:
1850:
1846:
1840:
1836:
1832:
1828:
1824:
1820:
1817:
1813:
1809:
1808:
1803:
1799:
1796:
1792:
1788:
1782:
1778:
1774:
1773:
1768:
1764:
1761:
1757:
1752:
1747:
1743:
1739:
1735:
1731:
1726:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1694:
1690:
1686:
1682:
1677:
1672:
1668:
1664:
1660:
1656:
1651:
1646:
1642:
1638:
1635:(2): 92–100,
1634:
1630:
1629:
1624:
1620:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1580:
1575:
1571:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1521:
1517:
1513:
1509:
1504:
1499:
1495:
1491:
1490:
1485:
1481:
1478:
1474:
1473:
1468:
1463:
1460:
1456:
1452:
1446:
1442:
1438:
1437:
1432:
1428:
1427:
1421:
1419:
1415:
1411:
1407:
1403:
1399:
1394:
1392:
1387:
1382:
1378:
1371:
1365:
1361:
1353:
1349:
1344:
1340:
1334:
1330:
1323:
1319:
1315:
1311:
1307:
1303:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1261:
1259:
1255:
1251:
1237:
1233:
1228:
1224:
1217:
1213:
1206:
1199:
1195:
1191:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1149:
1145:
1141:
1137:
1133:
1127:
1123:
1118:
1117:
1116:
1114:
1110:
1106:
1102:
1086:
1083:
1080:
1077:
1065:
1061:
1058:
1046:
1042:
1039:
1036:
1028:
1025:The image of
1020:
1016:
1011:} : {1})
1010:
1004:
1000:
997:
994:
990:
980:
976:
970:
966:
962:
958:
954:
950:
944:
940:
936:
929:
924:
920:
916:
912:
909:
905:
901:
897:
892:
888:
882:
878:
872:
868:
861:
857:
852:
848:
844:
840:
836:
835:
834:
832:
828:
823:
821:
817:
813:
809:
804:
800:
795:
791:
787:
783:
778:
774:
769:
765:
761:
756:
754:
750:
745:
741:
736:
732:
728:
724:
721:Two elements
719:
717:
705:
701:
682:
679:
676:
665:
661:
645:
637:
633:
626:
623:
620:
612:
608:
601:
598:
595:
587:
583:
576:
573:
570:
564:
558:
555:
548:
547:
546:
545:
541:
523:
519:
515:
512:
504:
500:
496:
492:
491:division ring
488:
484:
480:
477:
473:
470:
452:
449:
446:
435:
418:
415:
412:
409:
397:
393:
390:
378:
374:
371:
368:
357:
356:
347:
344:
341:
335:
331:
324:
320:
315:
311:
307:
303:
296:
292:
287:
283:
277:
273:
268:
264:
245:
242:
237:
233:
224:
220:
216:
213:
210:
198:
194:
188:
185:
182:
178:
163:
162:
160:
157:
153:
150:
147:
143:
140:
139:
138:
136:
133:
128:
119:
117:
113:
94:
91:
88:
58:
55:
52:
49:
46:
43:
35:
31:
27:
23:
19:
1869:
1834:
1830:
1806:
1771:
1707:
1703:
1632:
1626:
1583:
1577:
1534:
1530:
1493:
1487:
1470:
1435:
1413:
1409:
1405:
1401:
1397:
1395:
1388:
1380:
1376:
1369:
1363:
1359:
1351:
1347:
1342:
1338:
1332:
1328:
1321:
1317:
1313:
1309:
1305:
1301:
1296:
1292:
1291:matrix ring
1288:
1284:
1280:
1272:
1268:
1262:
1247:
1235:
1231:
1222:
1215:
1211:
1204:
1197:
1193:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1143:
1139:
1135:
1131:
1125:
1121:
1112:
1108:
1104:
1100:
1026:
1024:
1018:
1014:
1008:
1002:
998:
992:
988:
978:
974:
968:
964:
960:
952:
948:
942:
938:
934:
927:
922:
918:
914:
907:
903:
899:
895:
890:
886:
880:
876:
870:
866:
859:
855:
850:
846:
842:
838:
830:
826:
824:
819:
815:
811:
807:
802:
798:
793:
789:
785:
781:
776:
772:
767:
763:
759:
757:
752:
743:
739:
734:
730:
726:
722:
720:
713:
703:
540:direct limit
502:
498:
494:
482:
345:
339:
333:
329:
322:
318:
313:
309:
305:
294:
290:
285:
281:
275:
271:
267:directed set
262:
151:
141:
134:
129:
125:
17:
15:
735:perspective
733:are called
26:von Neumann
1889:Categories
1734:62.0648.03
1693:0014.22307
1424:References
1336:such that
1254:isomorphic
932:such that
906:copies of
737:, written
122:Definition
1853:0065-9266
1769:(1998) ,
1659:0027-8424
1600:0003-486X
1567:122594481
1551:0167-8094
1512:0008-4395
1477:EMS Press
1433:(1979) ,
1418:rank ring
1320:elements
1081:…
1013:, where {
766:, ... of
710:Dimension
646:⋯
621:⊂
596:⊂
571:⊂
516:⊗
413:…
243:∨
238:α
225:α
221:⋀
211:∨
199:α
186:∈
183:α
179:⋀
56:…
1760:16588050
1685:16588062
1375:∨ ... ∨
977: :
353:Examples
261:, where
156:complete
1879:0166126
1861:0634656
1837:(252),
1816:0157874
1795:0120174
1751:1076713
1712:Bibcode
1676:1076712
1637:Bibcode
1616:0088476
1608:1969811
1559:1554221
1520:0123923
1459:0598630
1327:, ...,
1138:) <
269:and if
146:modular
132:lattice
28: (
1877:
1859:
1851:
1841:
1814:
1793:
1783:
1758:
1748:
1740:
1732:
1691:
1683:
1673:
1665:
1657:
1614:
1606:
1598:
1565:
1557:
1549:
1518:
1510:
1457:
1447:
1367:, and
1209:, and
1021:and 1.
1742:86391
1738:JSTOR
1667:86390
1663:JSTOR
1604:JSTOR
1563:S2CID
1531:Order
1238:) ≤ 1
1218:) = 1
1200:) = 0
1129:then
1124:<
967:with
951:<
946:with
921:with
853:with
829:from
796:with
487:field
316:with
289:<
279:then
274:<
265:is a
1849:ISSN
1839:ISBN
1781:ISBN
1756:PMID
1681:PMID
1655:ISSN
1596:ISSN
1547:ISSN
1508:ISSN
1445:ISBN
1230:0 ≤
1181:) +
1173:) =
1161:) +
963:and
917:and
849:and
841:and
814:and
788:and
725:and
489:(or
34:1998
30:1936
1746:PMC
1730:JFM
1720:doi
1689:Zbl
1671:PMC
1645:doi
1588:doi
1539:doi
1498:doi
1384:= 1
1357:if
1355:= 0
1287:by
1225:= 1
1207:= 0
1119:If
987:as
930:≥ 0
862:= 0
822:.)
818:in
810:in
792:in
784:in
751:on
729:of
481:If
336:= 1
325:= 0
154:is
144:is
1891::
1875:MR
1857:MR
1855:,
1847:,
1835:34
1833:,
1829:,
1812:MR
1791:MR
1789:,
1779:,
1754:,
1744:,
1736:,
1728:,
1718:,
1708:22
1706:,
1687:,
1679:,
1669:,
1661:,
1653:,
1643:,
1633:22
1631:,
1612:MR
1610:,
1602:,
1594:,
1584:61
1561:,
1555:MR
1553:,
1545:,
1533:,
1516:MR
1514:,
1506:,
1492:,
1475:,
1469:,
1455:MR
1453:,
1443:,
1420:.
1362:≠
1346:∧
1169:∧
1157:∨
1007:({
985:/
941:+
939:nA
937:=
900:nA
889:+
879:∨
869:+
858:∧
801:≤
775:≤
762:,
742:∼
542:of
493:)
332:∨
327:,
321:∧
118:.
32:,
1722::
1714::
1647::
1639::
1590::
1541::
1535:1
1500::
1494:3
1414:R
1410:L
1406:L
1402:L
1398:R
1381:n
1377:a
1373:1
1370:a
1364:j
1360:i
1352:j
1348:a
1343:i
1339:a
1333:n
1329:a
1325:1
1322:a
1318:n
1314:n
1310:n
1306:R
1302:R
1300:(
1297:n
1293:M
1289:n
1285:n
1281:n
1273:L
1269:L
1236:a
1234:(
1232:D
1223:a
1216:a
1214:(
1212:D
1205:a
1198:a
1196:(
1194:D
1189:)
1187:b
1185:(
1183:D
1179:a
1177:(
1175:D
1171:b
1167:a
1165:(
1163:D
1159:b
1155:a
1153:(
1151:D
1146:)
1144:b
1142:(
1140:D
1136:a
1134:(
1132:D
1126:b
1122:a
1113:D
1109:D
1105:L
1101:n
1087:1
1084:,
1078:,
1072:n
1066:/
1062:2
1059:,
1053:n
1047:/
1043:1
1040:,
1037:0
1027:D
1019:a
1015:a
1009:a
1003:a
1001:(
999:D
993:C
989:C
981:)
979:A
975:B
973:(
969:A
965:B
961:A
956:.
953:B
949:C
943:C
935:B
928:n
923:A
919:B
915:A
908:A
904:n
896:n
891:B
887:A
881:b
877:a
871:B
867:A
860:b
856:a
851:b
847:a
843:B
839:A
831:L
827:D
820:B
816:b
812:A
808:a
803:b
799:a
794:B
790:b
786:A
782:a
777:B
773:A
768:L
764:B
760:A
753:L
744:b
740:a
731:L
727:b
723:a
704:F
686:]
683:1
680:,
677:0
674:[
643:)
638:8
634:F
630:(
627:G
624:P
618:)
613:4
609:F
605:(
602:G
599:P
593:)
588:2
584:F
580:(
577:G
574:P
568:)
565:F
562:(
559:G
556:P
524:2
520:F
513:V
503:V
499:V
495:F
483:V
468:.
456:]
453:1
450:,
447:0
444:[
422:}
419:1
416:,
410:,
404:n
398:/
394:2
391:,
385:n
379:/
375:1
372:,
369:0
366:{
346:L
342:.
340:L
334:b
330:a
323:b
319:a
314:b
310:a
306:L
295:β
291:a
286:α
282:a
276:β
272:α
263:A
249:)
246:b
234:a
230:(
217:=
214:b
206:)
195:a
189:A
173:(
158:.
152:L
148:.
142:L
135:L
98:]
95:1
92:,
89:0
86:[
64:n
59:,
53:,
50:1
47:,
44:0
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