1248:
1170:
1090:
867:
825:
783:
1368:
1344:
1014:
982:
931:
907:
417:
109:
698:= Ø. This remark holds more generally in any semigroup with zero. Furthermore, if every element has a unique pseudoinverse, then the semigroup is a
666:
The existence of a unique pseudoinverse implies the existence of a unique inverse, but the opposite is not true. For example, in the
57:. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via
1838:
1812:
1639:
1590:
1856:
1606:
1976:
1804:
1520:
1491:
1221:
1143:
1063:
840:
798:
756:
667:
70:
98:
388:(idempotent semigroup) is regular in the sense of this article, though this is not what is meant by a
1971:
1476:
389:
93:. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to
1349:
1325:
995:
963:
912:
888:
403:
73:
in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which
316:). Thus, another way of expressing definition (2) above is to say that in a regular semigroup,
1629:
1580:
1482:
All inverse semigroups are orthodox and locally inverse. The converse statements do not hold.
1822:
410:
1922:
747:
90:
74:
58:
8:
1530:
731:
699:
378:
1926:
1945:
1910:
1890:
1790:
1472:
1408:
396:
385:
82:
1950:
1852:
1834:
1808:
1635:
1586:
1525:
1496:
1319:
660:
430:
89:. It was Green's study of regular semigroups which led him to define his celebrated
1631:
Power
Algebras over Semirings: With Applications in Mathematics and Computer Science
1582:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
1940:
1930:
1880:
1872:
746:
automatically belongs to these ideals, without recourse to adjoining an identity.
86:
1830:
1798:
1479:
of the class of locally inverse semigroups and the class of orthodox semigroups.
711:
108:(French: demi-groupe inversif) was historically used as synonym in the papers of
670:, the empty transformation Ø does not have a unique pseudoinverse, because Ø = Ø
1794:
702:, and the unique pseudoinverse of an element coincides with the group inverse.
1885:
1965:
1515:
1849:
Monoids, Acts and
Categories with Applications to Wreath Products and Graphs
441:
be a regular semigroup in which idempotents commute. Then every element of
1954:
1935:
1903:
Proceedings of the
International Conference on Algebra and Its Applications
113:
1851:, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000,
429:
A regular semigroup in which idempotents commute (with idempotents) is an
1894:
1614:
1453:
1439:
1421:
1398:
934:
349:
302:
124:
There are two equivalent ways in which to define a regular semigroup
94:
21:
1876:
194:
To see the equivalence of these definitions, first suppose that
750:
can therefore be redefined for regular semigroups as follows:
659:
is shown to be unique. Conversely, it can be shown that any
424:
81:
in a semigroup was adapted from an analogous condition for
1915:
Proceedings of the
National Academy of Sciences of the USA
1373:
678:. The inverse of Ø is unique however, because only one
297:
The set of inverses (in the above sense) of an element
1863:
J. A. Green (1951). "On the structure of semigroups".
1634:. Springer Science & Business Media. p. 104.
1352:
1328:
1224:
1146:
1066:
998:
966:
915:
891:
843:
801:
759:
663:
is a regular semigroup in which idempotents commute.
1578:
1901:J. M. Howie, Semigroups, past, present and future,
730:belongs to the principal right, left and two-sided
116:) in the 1950s, and it is still used occasionally.
1362:
1338:
1242:
1164:
1084:
1008:
976:
925:
901:
861:
819:
777:
1627:
371:
1963:
1378:Some special classes of regular semigroups are:
1789:
1738:
1687:
1663:
1908:
1585:. American Mathematical Society. p. 181.
734:which it generates. In a regular semigroup
1862:
1572:
639:So, by commuting the pairs of idempotents
1944:
1934:
1884:
1475:of generalised inverse semigroups is the
1236:
1228:
1158:
1150:
1078:
1070:
855:
847:
813:
805:
771:
763:
682:satisfies the additional constraint that
425:Unique inverses and unique pseudoinverses
1621:
445:has at least one inverse. Suppose that
433:, or equivalently, every element has a
1964:
1654:Klip, Knauer and Mikhalev : p. 33
69:Regular semigroups were introduced by
1821:
1774:
1762:
1750:
1726:
1714:
1710:
1708:
1699:
1675:
1579:Christopher Hollings (16 July 2014).
1548:
1374:Special classes of regular semigroups
1243:{\displaystyle a\,{\mathcal {H}}\,b}
1165:{\displaystyle a\,{\mathcal {R}}\,b}
1085:{\displaystyle a\,{\mathcal {L}}\,b}
862:{\displaystyle a\,{\mathcal {J}}\,b}
820:{\displaystyle a\,{\mathcal {R}}\,b}
778:{\displaystyle a\,{\mathcal {L}}\,b}
726:; this is to ensure that an element
705:
1847:M. Kilp, U. Knauer, A.V. Mikhalev,
13:
1800:The algebraic theory of semigroups
1705:
1485:
1397:is an inverse semigroup, for each
1355:
1331:
1231:
1153:
1073:
1001:
969:
918:
894:
850:
808:
766:
420:of a regular semigroup is regular.
14:
1988:
77:were introduced. The concept of
1827:Fundamentals of Semigroup Theory
724:semigroup with identity adjoined
1768:
1756:
1744:
1732:
1720:
1693:
1681:
1669:
1657:
1648:
1599:
1563:
1554:
1542:
1428:Generalised inverse semigroups
1363:{\displaystyle {\mathcal {R}}}
1339:{\displaystyle {\mathcal {L}}}
1322:, then the idempotent in each
1049:denote the set of inverses of
1009:{\displaystyle {\mathcal {R}}}
977:{\displaystyle {\mathcal {L}}}
926:{\displaystyle {\mathcal {R}}}
902:{\displaystyle {\mathcal {L}}}
372:Examples of regular semigroups
332:. The product of any element
1:
1805:American Mathematical Society
1536:
1521:Special classes of semigroups
1436:generalised inverse semigroup
933:-class contains at least one
404:full transformation semigroup
119:
1791:Clifford, Alfred Hoblitzelle
1492:eventually regular semigroup
1033:be a regular semigroup; let
7:
1739:Clifford & Preston 2010
1688:Clifford & Preston 2010
1664:Clifford & Preston 2010
1509:
668:symmetric inverse semigroup
437:inverse. To see this, let
10:
1993:
1783:
1628:Jonathan S. Golan (1999).
1442:form a normal band, i.e.,
1383:Locally inverse semigroups
64:
27:in which every element is
674:Ø for any transformation
512:are idempotents as above.
324:) is nonempty, for every
198:is defined by (2). Then
31:, i.e., for each element
718:are defined in terms of
210:is defined by (1), then
206:in (1). Conversely, if
85:, already considered by
39:there exists an element
1909:J. von Neumann (1936).
1795:Preston, Gordon Bamford
881:In a regular semigroup
381:is a regular semigroup.
202:serves as the required
1936:10.1073/pnas.22.12.707
1823:Howie, John Mackintosh
1430:: a regular semigroup
1412:: a regular semigroup
1385:: a regular semigroup
1364:
1340:
1244:
1166:
1086:
1010:
978:
927:
903:
863:
821:
779:
738:, however, an element
1865:Annals of Mathematics
1504:-inversive) semigroup
1424:forms a subsemigroup.
1365:
1341:
1245:
1167:
1087:
1011:
979:
928:
904:
864:
822:
780:
411:Rees matrix semigroup
1977:Algebraic structures
1350:
1326:
1222:
1144:
1064:
996:
964:
913:
889:
841:
799:
757:
174:, in the sense that
148:, which is called a
1927:1936PNAS...22..707V
1531:Generalized inverse
1409:Orthodox semigroups
952:is any inverse for
106:inversive semigroup
1911:"On regular rings"
1886:10338.dmlcz/100067
1370:-class is unique.
1360:
1336:
1240:
1162:
1082:
1006:
974:
941:is any element of
923:
899:
859:
817:
775:
397:bicyclic semigroup
214:is an inverse for
163:(2) every element
97:was first made by
16:In mathematics, a
1867:. Second Series.
1840:978-0-19-851194-6
1814:978-0-8218-0272-4
1753:Proposition 2.4.1
1641:978-0-7923-5834-3
1592:978-1-4704-1493-1
1569:von Neumann 1936.
1526:Nambooripad order
1420:if its subset of
1320:inverse semigroup
869:if, and only if,
827:if, and only if,
785:if, and only if,
748:Green's relations
706:Green's relations
661:inverse semigroup
655:, the inverse of
453:has two inverses
431:inverse semigroup
418:homomorphic image
167:has at least one
75:Green's relations
59:Green's relations
18:regular semigroup
1984:
1972:Semigroup theory
1958:
1948:
1938:
1898:
1888:
1844:
1829:(1st ed.).
1818:
1778:
1772:
1766:
1760:
1754:
1748:
1742:
1736:
1730:
1724:
1718:
1712:
1703:
1697:
1691:
1685:
1679:
1673:
1667:
1661:
1655:
1652:
1646:
1645:
1625:
1619:
1618:
1613:. Archived from
1603:
1597:
1596:
1576:
1570:
1567:
1561:
1558:
1552:
1546:
1451:
1369:
1367:
1366:
1361:
1359:
1358:
1345:
1343:
1342:
1337:
1335:
1334:
1308:
1301:
1293:
1285:
1270:
1255:
1250:iff there exist
1249:
1247:
1246:
1241:
1235:
1234:
1214:
1207:
1192:
1177:
1172:iff there exist
1171:
1169:
1168:
1163:
1157:
1156:
1135:
1127:
1112:
1097:
1092:iff there exist
1091:
1089:
1088:
1083:
1077:
1076:
1021:
1015:
1013:
1012:
1007:
1005:
1004:
989:
983:
981:
980:
975:
973:
972:
950:
932:
930:
929:
924:
922:
921:
908:
906:
905:
900:
898:
897:
868:
866:
865:
860:
854:
853:
826:
824:
823:
818:
812:
811:
784:
782:
781:
776:
770:
769:
712:principal ideals
710:Recall that the
301:in an arbitrary
110:Gabriel Thierrin
87:John von Neumann
56:
1992:
1991:
1987:
1986:
1985:
1983:
1982:
1981:
1962:
1961:
1921:(12): 707–713.
1877:10.2307/1969317
1841:
1831:Clarendon Press
1815:
1803:. Vol. 2.
1786:
1781:
1773:
1769:
1761:
1757:
1749:
1745:
1737:
1733:
1725:
1721:
1713:
1706:
1698:
1694:
1686:
1682:
1674:
1670:
1662:
1658:
1653:
1649:
1642:
1626:
1622:
1605:
1604:
1600:
1593:
1577:
1573:
1568:
1564:
1559:
1555:
1547:
1543:
1539:
1512:
1488:
1486:Generalizations
1443:
1391:locally inverse
1376:
1354:
1353:
1351:
1348:
1347:
1330:
1329:
1327:
1324:
1323:
1306:
1299:
1291:
1283:
1268:
1253:
1230:
1229:
1223:
1220:
1219:
1212:
1205:
1190:
1175:
1152:
1151:
1145:
1142:
1141:
1133:
1125:
1110:
1095:
1072:
1071:
1065:
1062:
1061:
1041:be elements of
1019:
1000:
999:
997:
994:
993:
987:
968:
967:
965:
962:
961:
948:
917:
916:
914:
911:
910:
893:
892:
890:
887:
886:
849:
848:
842:
839:
838:
807:
806:
800:
797:
796:
765:
764:
758:
755:
754:
714:of a semigroup
708:
427:
374:
122:
67:
48:
12:
11:
5:
1990:
1980:
1979:
1974:
1960:
1959:
1906:
1899:
1871:(1): 163–172.
1860:
1845:
1839:
1819:
1813:
1785:
1782:
1780:
1779:
1767:
1755:
1743:
1731:
1719:
1704:
1692:
1680:
1668:
1656:
1647:
1640:
1620:
1617:on 1999-11-04.
1611:www.csd.uwo.ca
1607:"Publications"
1598:
1591:
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1528:
1523:
1518:
1511:
1508:
1507:
1506:
1494:
1487:
1484:
1469:
1468:
1425:
1405:
1375:
1372:
1357:
1333:
1312:
1311:
1239:
1233:
1227:
1217:
1161:
1155:
1149:
1139:
1081:
1075:
1069:
1003:
971:
920:
896:
879:
878:
858:
852:
846:
836:
816:
810:
804:
794:
774:
768:
762:
707:
704:
637:
636:
514:
513:
426:
423:
422:
421:
414:
407:
400:
393:
382:
373:
370:
308:is denoted by
192:
191:
161:
140:, there is an
121:
118:
112:(a student of
66:
63:
9:
6:
4:
3:
2:
1989:
1978:
1975:
1973:
1970:
1969:
1967:
1956:
1952:
1947:
1942:
1937:
1932:
1928:
1924:
1920:
1916:
1912:
1907:
1905:, 2002, 6–20.
1904:
1900:
1896:
1892:
1887:
1882:
1878:
1874:
1870:
1866:
1861:
1858:
1857:3-11-015248-7
1854:
1850:
1846:
1842:
1836:
1832:
1828:
1824:
1820:
1816:
1810:
1806:
1802:
1801:
1796:
1792:
1788:
1787:
1776:
1771:
1764:
1759:
1752:
1747:
1740:
1735:
1728:
1723:
1717:Theorem 5.1.1
1716:
1711:
1709:
1701:
1696:
1689:
1684:
1677:
1672:
1665:
1660:
1651:
1643:
1637:
1633:
1632:
1624:
1616:
1612:
1608:
1602:
1594:
1588:
1584:
1583:
1575:
1566:
1557:
1550:
1545:
1541:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1516:Biordered set
1514:
1513:
1505:
1503:
1499:
1495:
1493:
1490:
1489:
1483:
1480:
1478:
1474:
1466:
1462:
1458:
1455:
1450:
1446:
1441:
1437:
1433:
1429:
1426:
1423:
1419:
1415:
1411:
1410:
1406:
1403:
1400:
1396:
1392:
1388:
1384:
1381:
1380:
1379:
1371:
1321:
1317:
1309:
1302:
1295:
1287:
1279:
1275:
1271:
1264:
1260:
1256:
1237:
1225:
1218:
1215:
1208:
1201:
1197:
1193:
1186:
1182:
1178:
1159:
1147:
1140:
1137:
1129:
1121:
1117:
1113:
1106:
1102:
1098:
1079:
1067:
1060:
1059:
1058:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1022:
991:
959:
955:
951:
944:
940:
936:
884:
876:
872:
856:
844:
837:
834:
830:
814:
802:
795:
792:
788:
772:
760:
753:
752:
751:
749:
745:
741:
737:
733:
729:
725:
721:
717:
713:
703:
701:
697:
693:
689:
685:
681:
677:
673:
669:
664:
662:
658:
654:
650:
646:
642:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
522:
519:
518:
517:
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
471:
467:
464:
463:
462:
460:
456:
452:
448:
444:
440:
436:
432:
419:
415:
412:
408:
405:
401:
398:
394:
391:
387:
383:
380:
376:
375:
369:
367:
363:
359:
355:
351:
347:
343:
339:
335:
331:
327:
323:
319:
315:
311:
307:
304:
300:
295:
293:
289:
285:
281:
277:
273:
269:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
189:
185:
181:
177:
173:
170:
166:
162:
159:
155:
151:
150:pseudoinverse
147:
143:
139:
135:
132:(1) for each
131:
130:
129:
127:
117:
115:
111:
107:
102:
100:
96:
92:
88:
84:
80:
76:
72:
62:
60:
55:
51:
46:
42:
38:
34:
30:
26:
23:
19:
1918:
1914:
1902:
1868:
1864:
1848:
1826:
1799:
1770:
1765:ch. 6, § 2.4
1758:
1746:
1734:
1722:
1695:
1683:
1671:
1659:
1650:
1630:
1623:
1615:the original
1610:
1601:
1581:
1574:
1565:
1556:
1544:
1501:
1500:-dense (aka
1497:
1481:
1477:intersection
1470:
1464:
1460:
1456:
1448:
1444:
1435:
1434:is called a
1431:
1427:
1417:
1413:
1407:
1401:
1394:
1390:
1386:
1382:
1377:
1315:
1313:
1304:
1297:
1289:
1281:
1280:) such that
1277:
1273:
1266:
1262:
1258:
1251:
1210:
1203:
1202:) such that
1199:
1195:
1188:
1184:
1180:
1173:
1131:
1123:
1122:) such that
1119:
1115:
1108:
1104:
1100:
1093:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1025:
1017:
1016:-related to
985:
984:-related to
957:
953:
946:
942:
938:
882:
880:
874:
870:
832:
828:
790:
786:
743:
739:
735:
727:
723:
719:
715:
709:
695:
691:
687:
683:
679:
675:
671:
665:
656:
652:
648:
644:
640:
638:
632:
628:
624:
620:
616:
612:
608:
604:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
515:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
465:
458:
454:
450:
446:
442:
438:
434:
428:
390:regular band
365:
361:
357:
353:
348:) is always
345:
341:
337:
333:
329:
325:
321:
317:
313:
309:
305:
298:
296:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
193:
187:
183:
179:
175:
171:
168:
164:
157:
153:
149:
145:
141:
137:
133:
125:
123:
114:Paul Dubreil
105:
103:
78:
68:
53:
49:
44:
40:
36:
32:
28:
24:
17:
15:
1702:Lemma 2.4.4
1666:Lemma 1.14.
1560:Howie 2002.
1454:idempotents
1440:idempotents
1422:idempotents
413:is regular.
406:is regular.
399:is regular.
71:J. A. Green
1966:Categories
1775:Howie 1995
1763:Howie 1995
1751:Howie 1995
1741:Lemma 1.13
1727:Howie 1995
1715:Howie 1995
1700:Howie 1995
1676:Howie 1995
1549:Howie 1995
1537:References
1399:idempotent
1045:, and let
935:idempotent
350:idempotent
120:The basics
99:David Rees
95:semigroups
79:regularity
47:such that
1797:(2010) .
694:, namely
336:with any
303:semigroup
104:The term
91:relations
22:semigroup
1955:16577757
1825:(1995).
1510:See also
1452:for all
1418:orthodox
1057:. Then
1027:Theorem.
885:, every
461:, i.e.,
360:, since
218:, since
1946:1076849
1923:Bibcode
1895:1969317
1784:Sources
1438:if its
956:, then
496:. Also
169:inverse
152:, with
65:History
29:regular
1953:
1943:
1893:
1855:
1837:
1811:
1777:p. 222
1638:
1589:
1346:- and
1318:is an
1265:) and
1187:) and
1107:) and
937:. If
909:- and
732:ideals
722:, the
651:&
643:&
435:unique
384:Every
377:Every
1891:JSTOR
1729:p. 55
1690:p. 26
1678:p. 52
1551:p. 54
1473:class
700:group
625:cabac
516:Then
379:group
246:and (
83:rings
20:is a
1951:PMID
1853:ISBN
1835:ISBN
1809:ISBN
1636:ISBN
1587:ISBN
1471:The
1449:xzyx
1445:xyzx
1296:and
1047:V(x)
1037:and
1029:Let
992:and
945:and
647:and
587:) =
575:) =
508:and
488:and
457:and
416:The
402:Any
395:The
386:band
354:abab
278:) =
270:) =
258:) =
238:) =
182:and
1941:PMC
1931:doi
1881:hdl
1873:doi
1416:is
1395:eSe
1393:if
1389:is
1314:If
1272:in
1257:in
1194:in
1179:in
1114:in
1099:in
1053:in
960:is
875:SbS
871:SaS
744:axa
629:cac
621:bac
617:aba
609:bac
597:bac
577:bac
565:bac
557:aca
553:bac
541:bac
533:aca
525:bab
490:cac
482:aca
474:bab
466:aba
449:in
362:aba
340:in
328:in
292:xax
284:axa
276:xax
268:xax
264:axa
256:xax
248:xax
240:axa
232:axa
224:xax
212:xax
184:bab
176:aba
154:axa
144:in
136:in
50:axa
43:in
35:in
1968::
1949:.
1939:.
1929:.
1919:22
1917:.
1913:.
1889:.
1879:.
1869:54
1833:.
1807:.
1793:;
1707:^
1609:.
1463:,
1459:,
1447:=
1305:bb
1303:=
1298:aa
1288:=
1211:bb
1209:=
1204:aa
1130:=
1023:.
1018:aa
873:=
833:bS
831:=
829:aS
791:Sb
789:=
787:Sa
742:=
686:=
653:ca
649:ba
645:ac
641:ab
631:=
627:=
623:=
611:=
605:ba
601:ca
599:=
593:ca
589:ba
585:ac
583:)(
581:ab
573:ab
571:)(
569:ac
563:=
551:=
539:=
527:=
523:=
510:ca
506:ac
504:,
502:ba
500:,
498:ab
492:=
484:=
480:,
476:=
472:,
468:=
409:A
368:.
364:=
358:ab
356:=
352::
294:.
290:=
272:xa
266:)(
242:=
236:xa
230:=
186:=
178:=
156:=
128::
101:.
61:.
52:=
1957:.
1933::
1925::
1897:.
1883::
1875::
1859:.
1843:.
1817:.
1644:.
1595:.
1502:E
1498:E
1467:.
1465:z
1461:y
1457:x
1432:S
1414:S
1404:.
1402:e
1387:S
1356:R
1332:L
1316:S
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1307:′
1300:′
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1274:V
1269:′
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1263:a
1261:(
1259:V
1254:′
1252:a
1238:b
1232:H
1226:a
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1213:′
1206:′
1200:b
1198:(
1196:V
1191:′
1189:b
1185:a
1183:(
1181:V
1176:′
1174:a
1160:b
1154:R
1148:a
1138:;
1136:b
1134:′
1132:b
1128:a
1126:′
1124:a
1120:b
1118:(
1116:V
1111:′
1109:b
1105:a
1103:(
1101:V
1096:′
1094:a
1080:b
1074:L
1068:a
1055:S
1051:x
1043:S
1039:b
1035:a
1031:S
1020:′
1002:R
990:a
988:′
986:a
970:L
958:a
954:a
949:′
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943:S
939:a
919:R
895:L
883:S
877:.
857:b
851:J
845:a
835:;
815:b
809:R
803:a
793:;
773:b
767:L
761:a
740:a
736:S
728:a
720:S
716:S
696:f
692:f
690:Ø
688:f
684:f
680:f
676:f
672:f
657:a
635:.
633:c
619:)
615:(
613:c
607:)
603:(
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567:(
561:b
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543:(
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447:a
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439:S
392:.
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344:(
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338:b
334:a
330:S
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320:(
318:V
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312:(
310:V
306:S
299:a
288:x
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282:(
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190:.
188:b
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160:;
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146:S
142:x
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134:a
126:S
54:a
45:S
41:x
37:S
33:a
25:S
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