230:
1030:
480:
1195:
859:
1230:
1065:
894:
729:
694:
1093:
517:
922:
757:
547:
1150:
979:
814:
655:
598:
406:
355:
108:
88:
1338:
519:
there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970):
1564:
278:
can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether
113:
1540:
1499:
1465:
17:
1559:
1457:
298:), since these are equal. Exponentiation to the power of zero can also be defined if the operation has an
110:
by itself several times, it doesn't matter in which order the operations are carried out, so for instance
1491:
984:
411:
310:
1155:
819:
1532:
1526:
1200:
1035:
864:
699:
660:
1329:
1072:
496:
1509:
1424:
1377:
59:
1517:
1475:
1432:
1385:
8:
1441:
901:
736:
526:
306:
245:
241:
1098:
927:
762:
603:
552:
360:
316:
1393:
Gainov, A. T. (1970). "Power-associative algebras over a finite-characteristic field".
1365:
93:
73:
1536:
1495:
1461:
1412:
1395:
1357:
1239:
power-associative algebras with unit, which basically asserts that multiplication of
1513:
1471:
1428:
1404:
1381:
1347:
299:
275:
253:
70:
generated by any element is associative. Concretely, this means that if an element
43:
35:
1505:
1420:
1373:
302:, so the existence of identity elements is useful in power-associative contexts.
63:
271:
1263:) to be the element of the algebra resulting from the obvious substitution of
1553:
1416:
1361:
1316:
261:
47:
1483:
1453:
490:
1490:. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2.
1445:
1236:
31:
1408:
1369:
1240:
483:
265:
67:
1488:
Introduction to octonion and other non-associative algebras in physics
1352:
1333:
257:
249:
313:
0, an algebra is power-associative if and only if it satisfies
1452:. Colloquium Publications. Vol. 44. With a preface by
1439:
252:, which are non-associative) and even non-alternative
1203:
1158:
1101:
1075:
1038:
987:
930:
904:
867:
822:
765:
739:
702:
663:
606:
555:
529:
499:
414:
363:
319:
116:
96:
76:
225:{\displaystyle x*(x*(x*x))=(x*(x*x))*x=(x*x)*(x*x)}
1224:
1189:
1144:
1087:
1059:
1024:
973:
916:
888:
853:
808:
751:
723:
688:
649:
592:
541:
511:
474:
400:
349:
224:
102:
82:
1339:Transactions of the American Mathematical Society
1551:
1528:An introduction to non-associative algebras
244:is power-associative, but so are all other
235:
1351:
66:) is said to be power-associative if the
1524:
14:
1552:
1392:
1328:
1482:
1271:. Then for any two such polynomials
24:
264:. Any algebra whose elements are
25:
1576:
1565:Properties of binary operations
1448:; Tignol, Jean-Pierre (1998).
1219:
1133:
1102:
1054:
962:
931:
883:
797:
766:
718:
638:
607:
581:
556:
469:
460:
448:
439:
433:
415:
389:
364:
338:
320:
219:
207:
201:
189:
177:
174:
162:
153:
147:
144:
132:
123:
27:Property of a binary operation
13:
1:
1458:American Mathematical Society
1322:
1235:A substitution law holds for
1025:{\displaystyle n=3,4,6,5^{k}}
475:{\displaystyle :=(xy)z-x(yz)}
53:
7:
1310:
1190:{\displaystyle n=3,4,p^{k}}
854:{\displaystyle n=4,5,3^{k}}
268:is also power-associative.
10:
1581:
1492:Cambridge University Press
1255:in such an algebra define
489:Over an infinite field of
90:is performed an operation
1334:"Power-associative rings"
1225:{\displaystyle k=1,2...)}
1060:{\displaystyle k=1,2...)}
889:{\displaystyle k=1,2...)}
724:{\displaystyle k=2,3...)}
689:{\displaystyle n=3,2^{k}}
1525:Schafer, R. D. (1995) .
1560:Non-associative algebra
1450:The book of involutions
1243:works as expected. For
236:Examples and properties
46:that is a weak form of
1226:
1191:
1146:
1089:
1088:{\displaystyle p>5}
1061:
1026:
975:
918:
890:
855:
810:
753:
725:
690:
651:
594:
543:
513:
512:{\displaystyle p>0}
476:
402:
351:
282:should be defined as (
226:
104:
84:
1247:a real polynomial in
1227:
1192:
1147:
1090:
1062:
1027:
976:
919:
891:
856:
811:
754:
726:
691:
652:
595:
544:
514:
477:
403:
352:
227:
105:
85:
62:(or more generally a
1442:Merkurjev, Alexander
1201:
1156:
1099:
1073:
1036:
985:
928:
902:
865:
820:
763:
737:
700:
661:
604:
553:
527:
497:
412:
361:
317:
274:to the power of any
246:alternative algebras
114:
94:
74:
917:{\displaystyle p=5}
752:{\displaystyle p=3}
542:{\displaystyle p=2}
242:associative algebra
42:is a property of a
40:power associativity
1531:. Dover. pp.
1456:. Providence, RI:
1440:Knus, Max-Albert;
1409:10.1007/BF02219846
1222:
1187:
1145:{\displaystyle =0}
1142:
1085:
1057:
1022:
974:{\displaystyle =0}
971:
914:
886:
851:
809:{\displaystyle =0}
806:
749:
721:
686:
650:{\displaystyle =0}
647:
593:{\displaystyle =0}
590:
539:
509:
472:
401:{\displaystyle =0}
398:
350:{\displaystyle =0}
347:
222:
100:
80:
34:, specifically in
1396:Algebra and Logic
1330:Albert, A. Adrian
254:flexible algebras
103:{\displaystyle *}
83:{\displaystyle x}
18:Power associative
16:(Redirected from
1572:
1546:
1521:
1479:
1436:
1389:
1355:
1306:
1231:
1229:
1228:
1223:
1196:
1194:
1193:
1188:
1186:
1185:
1151:
1149:
1148:
1143:
1120:
1119:
1094:
1092:
1091:
1086:
1066:
1064:
1063:
1058:
1031:
1029:
1028:
1023:
1021:
1020:
980:
978:
977:
972:
949:
948:
923:
921:
920:
915:
895:
893:
892:
887:
860:
858:
857:
852:
850:
849:
815:
813:
812:
807:
784:
783:
758:
756:
755:
750:
730:
728:
727:
722:
695:
693:
692:
687:
685:
684:
656:
654:
653:
648:
625:
624:
599:
597:
596:
591:
574:
573:
548:
546:
545:
540:
518:
516:
515:
510:
481:
479:
478:
473:
407:
405:
404:
399:
376:
375:
356:
354:
353:
348:
300:identity element
276:positive integer
231:
229:
228:
223:
109:
107:
106:
101:
89:
87:
86:
81:
44:binary operation
36:abstract algebra
21:
1580:
1579:
1575:
1574:
1573:
1571:
1570:
1569:
1550:
1549:
1543:
1502:
1468:
1353:10.2307/1990399
1325:
1313:
1280:
1279:, we have that
1202:
1199:
1198:
1181:
1177:
1157:
1154:
1153:
1109:
1105:
1100:
1097:
1096:
1074:
1071:
1070:
1037:
1034:
1033:
1016:
1012:
986:
983:
982:
938:
934:
929:
926:
925:
903:
900:
899:
866:
863:
862:
845:
841:
821:
818:
817:
773:
769:
764:
761:
760:
738:
735:
734:
701:
698:
697:
680:
676:
662:
659:
658:
614:
610:
605:
602:
601:
569:
565:
554:
551:
550:
528:
525:
524:
498:
495:
494:
493:characteristic
486:(Albert 1948).
413:
410:
409:
371:
367:
362:
359:
358:
318:
315:
314:
238:
115:
112:
111:
95:
92:
91:
75:
72:
71:
56:
28:
23:
22:
15:
12:
11:
5:
1578:
1568:
1567:
1562:
1548:
1547:
1541:
1522:
1500:
1494:. p. 17.
1480:
1466:
1437:
1390:
1324:
1321:
1320:
1319:
1312:
1309:
1251:, and for any
1233:
1232:
1221:
1218:
1215:
1212:
1209:
1206:
1184:
1180:
1176:
1173:
1170:
1167:
1164:
1161:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1118:
1115:
1112:
1108:
1104:
1084:
1081:
1078:
1067:
1056:
1053:
1050:
1047:
1044:
1041:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
970:
967:
964:
961:
958:
955:
952:
947:
944:
941:
937:
933:
913:
910:
907:
896:
885:
882:
879:
876:
873:
870:
848:
844:
840:
837:
834:
831:
828:
825:
805:
802:
799:
796:
793:
790:
787:
782:
779:
776:
772:
768:
748:
745:
742:
731:
720:
717:
714:
711:
708:
705:
683:
679:
675:
672:
669:
666:
646:
643:
640:
637:
634:
631:
628:
623:
620:
617:
613:
609:
589:
586:
583:
580:
577:
572:
568:
564:
561:
558:
538:
535:
532:
508:
505:
502:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
397:
394:
391:
388:
385:
382:
379:
374:
370:
366:
346:
343:
340:
337:
334:
331:
328:
325:
322:
311:characteristic
272:Exponentiation
262:Okubo algebras
237:
234:
221:
218:
215:
212:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
99:
79:
55:
52:
26:
9:
6:
4:
3:
2:
1577:
1566:
1563:
1561:
1558:
1557:
1555:
1544:
1542:0-486-68813-5
1538:
1534:
1530:
1529:
1523:
1519:
1515:
1511:
1507:
1503:
1501:0-521-01792-0
1497:
1493:
1489:
1485:
1484:Okubo, Susumu
1481:
1477:
1473:
1469:
1467:0-8218-0904-0
1463:
1459:
1455:
1451:
1447:
1443:
1438:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1397:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1354:
1349:
1345:
1341:
1340:
1335:
1331:
1327:
1326:
1318:
1317:Alternativity
1315:
1314:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1216:
1213:
1210:
1207:
1204:
1182:
1178:
1174:
1171:
1168:
1165:
1162:
1159:
1139:
1136:
1130:
1127:
1124:
1121:
1116:
1113:
1110:
1106:
1082:
1079:
1076:
1068:
1051:
1048:
1045:
1042:
1039:
1017:
1013:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
968:
965:
959:
956:
953:
950:
945:
942:
939:
935:
911:
908:
905:
897:
880:
877:
874:
871:
868:
846:
842:
838:
835:
832:
829:
826:
823:
803:
800:
794:
791:
788:
785:
780:
777:
774:
770:
746:
743:
740:
732:
715:
712:
709:
706:
703:
681:
677:
673:
670:
667:
664:
644:
641:
635:
632:
629:
626:
621:
618:
615:
611:
587:
584:
578:
575:
570:
566:
562:
559:
536:
533:
530:
522:
521:
520:
506:
503:
500:
492:
487:
485:
466:
463:
457:
454:
451:
445:
442:
436:
430:
427:
424:
421:
418:
395:
392:
386:
383:
380:
377:
372:
368:
344:
341:
335:
332:
329:
326:
323:
312:
308:
303:
301:
297:
293:
289:
285:
281:
277:
273:
269:
267:
263:
259:
255:
251:
247:
243:
233:
216:
213:
210:
204:
198:
195:
192:
186:
183:
180:
171:
168:
165:
159:
156:
150:
141:
138:
135:
129:
126:
120:
117:
97:
77:
69:
65:
61:
51:
49:
48:associativity
45:
41:
37:
33:
19:
1527:
1487:
1454:Jacques Tits
1449:
1446:Rost, Markus
1400:
1394:
1343:
1337:
1302:
1298:
1294:
1290:
1286:
1282:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1234:
488:
304:
295:
291:
287:
283:
279:
270:
239:
57:
39:
29:
1403:(1): 5–19.
1346:: 552–593.
1241:polynomials
32:mathematics
1554:Categories
1518:0841.17001
1476:0955.16001
1433:0208.04001
1386:0033.15402
1323:References
484:associator
266:idempotent
248:(like the
68:subalgebra
54:Definition
1417:0002-9947
1362:0002-9947
1114:−
943:−
778:−
619:−
455:−
258:sedenions
256:like the
250:octonions
214:∗
205:∗
196:∗
181:∗
169:∗
160:∗
139:∗
130:∗
121:∗
98:∗
1486:(1995).
1332:(1948).
1311:See also
408:, where
1533:128–148
1510:1356224
1425:0281764
1378:0027750
1370:1990399
482:is the
305:Over a
60:algebra
1539:
1516:
1508:
1498:
1474:
1464:
1431:
1423:
1415:
1384:
1376:
1368:
1360:
290:or as
240:Every
1366:JSTOR
1267:into
491:prime
307:field
64:magma
1537:ISBN
1496:ISBN
1462:ISBN
1413:ISSN
1358:ISSN
1289:) =
1275:and
1237:real
1217:2...
1152:for
1080:>
1069:For
1052:2...
981:for
898:For
881:2...
816:for
733:For
716:3...
657:for
600:and
523:For
504:>
357:and
260:and
1514:Zbl
1472:Zbl
1429:Zbl
1405:doi
1382:Zbl
1348:doi
309:of
58:An
30:In
1556::
1535:.
1512:.
1506:MR
1504:.
1470:.
1460:.
1444:;
1427:.
1421:MR
1419:.
1411:.
1399:.
1380:.
1374:MR
1372:.
1364:.
1356:.
1344:64
1342:.
1336:.
1307:.
1285:)(
1283:fg
1095::
924::
759::
549::
437::=
296:xx
284:xx
232:.
50:.
38:,
1545:.
1520:.
1478:.
1435:.
1407::
1401:9
1388:.
1350::
1305:)
1303:a
1301:(
1299:g
1297:)
1295:a
1293:(
1291:f
1287:a
1281:(
1277:g
1273:f
1269:f
1265:a
1261:a
1259:(
1257:f
1253:a
1249:x
1245:f
1220:)
1214:,
1211:1
1208:=
1205:k
1197:(
1183:k
1179:p
1175:,
1172:4
1169:,
1166:3
1163:=
1160:n
1140:0
1137:=
1134:]
1131:x
1128:,
1125:x
1122:,
1117:2
1111:n
1107:x
1103:[
1083:5
1077:p
1055:)
1049:,
1046:1
1043:=
1040:k
1032:(
1018:k
1014:5
1010:,
1007:6
1004:,
1001:4
998:,
995:3
992:=
989:n
969:0
966:=
963:]
960:x
957:,
954:x
951:,
946:2
940:n
936:x
932:[
912:5
909:=
906:p
884:)
878:,
875:1
872:=
869:k
861:(
847:k
843:3
839:,
836:5
833:,
830:4
827:=
824:n
804:0
801:=
798:]
795:x
792:,
789:x
786:,
781:2
775:n
771:x
767:[
747:3
744:=
741:p
719:)
713:,
710:2
707:=
704:k
696:(
682:k
678:2
674:,
671:3
668:=
665:n
645:0
642:=
639:]
636:x
633:,
630:x
627:,
622:2
616:n
612:x
608:[
588:0
585:=
582:]
579:x
576:,
571:2
567:x
563:,
560:x
557:[
537:2
534:=
531:p
507:0
501:p
470:)
467:z
464:y
461:(
458:x
452:z
449:)
446:y
443:x
440:(
434:]
431:z
428:,
425:y
422:,
419:x
416:[
396:0
393:=
390:]
387:x
384:,
381:x
378:,
373:2
369:x
365:[
345:0
342:=
339:]
336:x
333:,
330:x
327:,
324:x
321:[
294:(
292:x
288:x
286:)
280:x
220:)
217:x
211:x
208:(
202:)
199:x
193:x
190:(
187:=
184:x
178:)
175:)
172:x
166:x
163:(
157:x
154:(
151:=
148:)
145:)
142:x
136:x
133:(
127:x
124:(
118:x
78:x
20:)
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