986:
316:
887:
840:
696:
768:
1008:
639:
519:
493:
399:
796:
339:
232:
209:
1072:
1052:
1028:
880:
860:
740:
720:
663:
619:
599:
579:
559:
539:
473:
419:
379:
359:
252:
186:
162:
142:
1956:
1907:
1819:
1879:
1899:
1925:
257:
1935:
1811:
1735:
1930:
1727:
1746:
by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.
110:
37:
981:{\displaystyle s*t={\begin{cases}st&{\text{if }}s,t,st\in S\setminus I\\0&{\text{otherwise}}.\end{cases}}}
1988:
801:
1743:
1863:
1739:
1972:
642:
45:
908:
1731:
1799:
1780:
666:
671:
1952:
1903:
1875:
1815:
1851:
745:
1847:
1772:
21:
1829:
993:
624:
498:
478:
384:
1871:
1825:
773:
321:
214:
191:
1803:
1057:
1037:
1013:
865:
845:
725:
705:
648:
604:
584:
564:
544:
524:
458:
404:
364:
344:
237:
171:
147:
127:
1982:
1784:
1726:
Some of the cases that have been studied extensively include: ideal extensions of
17:
1968:
1776:
87:
retain their identity. The new semigroup obtained in this way is called the
56:
41:
122:
1518:} with the binary operation defined by the following Cayley table:
1106:} with the binary operation defined by the following Cayley table:
974:
311:{\displaystyle SI=\{sx\mid s\in S{\text{ and }}x\in I\}}
1967:
This article incorporates material from Rees factor on
109:
The concept of Rees factor semigroup was introduced by
1894:
Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter (2002).
1060:
1040:
1016:
996:
890:
868:
848:
804:
776:
748:
728:
708:
674:
651:
627:
607:
587:
567:
547:
527:
501:
481:
461:
407:
387:
367:
347:
324:
260:
240:
217:
194:
174:
150:
130:
1949:
Inverse semigroups: the theory of partial symmetries
1844:
Inverse
Semigroups: the theory of partial symmetries
862:is a new element and the product (here denoted by
1066:
1046:
1022:
1002:
980:
874:
854:
834:
790:
762:
734:
714:
690:
657:
633:
613:
593:
573:
553:
533:
513:
487:
467:
413:
393:
373:
353:
333:
310:
246:
226:
203:
180:
156:
136:
1810:. Mathematical Surveys, No. 7. Providence, R.I.:
1973:Creative Commons Attribution/Share-Alike License
75:one can construct a new semigroup by collapsing
798:. The Rees factor semigroup has underlying set
8:
1695:is called an ideal extension of a semigroup
829:
823:
508:
502:
305:
270:
79:into a single element while the elements of
742:. For notational convenience the semigroup
1808:The algebraic theory of semigroups. Vol. I
1059:
1039:
1015:
995:
963:
919:
903:
889:
867:
847:
803:
780:
775:
752:
747:
727:
707:
683:
678:
673:
650:
626:
606:
586:
566:
546:
526:
500:
480:
460:
406:
386:
366:
346:
323:
291:
259:
239:
216:
193:
173:
149:
129:
1520:
1108:
835:{\displaystyle (S\setminus I)\cup \{0\}}
1755:
948:
811:
44:constructed using a semigroup and an
7:
14:
1763:D. Rees (1940). "On semigroups".
1868:Fundamentals of Semigroup Theory
475:. The equivalence classes under
1896:The concise handbook of algebra
1486:. The Rees factor semigroup of
1030:as defined above is called the
1971:, which is licensed under the
1711:and the Rees factor semigroup
817:
805:
455:is an equivalence relation in
1:
1812:American Mathematical Society
1736:completely 0-simple semigroup
1800:Clifford, Alfred Hoblitzelle
1728:completely simple semigroups
1931:Encyclopedia of Mathematics
1926:"Extension of a semi-group"
361:be an ideal of a semigroup
2005:
1777:10.1017/S0305004100017436
691:{\displaystyle S/{\rho }}
89:Rees factor semigroup of
1924:Gluskin, L.M. (2001) ,
1804:Preston, Gordon Bamford
1355:} which is a subset of
1082:Consider the semigroup
763:{\displaystyle S/\rho }
698:is, by definition, the
495:are the singleton sets
30:Rees quotient semigroup
1068:
1048:
1024:
1004:
982:
876:
856:
836:
792:
764:
736:
716:
692:
659:
635:
615:
595:
575:
555:
535:
515:
489:
469:
415:
395:
375:
355:
335:
312:
248:
228:
205:
182:
158:
138:
46:ideal of the semigroup
1947:Lawson, M.V. (1998).
1765:Proc. Camb. Phil. Soc
1740:commutative semigroup
1069:
1049:
1025:
1005:
1003:{\displaystyle \rho }
983:
877:
857:
837:
793:
765:
737:
717:
700:Rees factor semigroup
693:
660:
636:
634:{\displaystyle \rho }
616:
596:
576:
556:
536:
516:
514:{\displaystyle \{x\}}
490:
488:{\displaystyle \rho }
470:
432: ⇔ either
416:
396:
394:{\displaystyle \rho }
376:
356:
336:
313:
249:
229:
206:
183:
159:
139:
26:Rees factor semigroup
1951:. World Scientific.
1914:(pp. 1–3)
1058:
1038:
1014:
994:
888:
866:
846:
802:
774:
746:
726:
706:
672:
649:
625:
605:
585:
565:
545:
525:
499:
479:
459:
405:
385:
365:
345:
322:
318:, and similarly for
258:
238:
215:
192:
172:
148:
128:
791:{\displaystyle S/I}
770:is also denoted as
98:and is denoted by
1064:
1044:
1020:
1000:
978:
973:
872:
852:
832:
788:
760:
732:
712:
688:
667:quotient semigroup
655:
631:
611:
591:
571:
551:
531:
511:
485:
465:
411:
391:
371:
351:
334:{\displaystyle IS}
331:
308:
244:
227:{\displaystyle IS}
224:
204:{\displaystyle SI}
201:
178:
154:
134:
1958:978-981-02-3316-7
1909:978-0-7923-7072-7
1852:Google Books link
1821:978-0-8218-0272-4
1719:is isomorphic to
1684:
1683:
1341:
1340:
1067:{\displaystyle I}
1047:{\displaystyle S}
1023:{\displaystyle S}
966:
922:
875:{\displaystyle *}
855:{\displaystyle 0}
735:{\displaystyle I}
715:{\displaystyle S}
658:{\displaystyle S}
614:{\displaystyle S}
594:{\displaystyle I}
574:{\displaystyle I}
554:{\displaystyle I}
534:{\displaystyle x}
468:{\displaystyle S}
414:{\displaystyle S}
374:{\displaystyle S}
354:{\displaystyle I}
294:
247:{\displaystyle I}
181:{\displaystyle S}
157:{\displaystyle S}
137:{\displaystyle I}
117:Formal definition
1996:
1989:Semigroup theory
1962:
1939:
1938:
1921:
1915:
1913:
1891:
1885:
1884:
1860:
1854:
1848:World Scientific
1840:
1834:
1833:
1796:
1790:
1788:
1760:
1521:
1109:
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
987:
985:
984:
979:
977:
976:
967:
964:
923:
920:
882:) is defined by
881:
879:
878:
873:
861:
859:
858:
853:
841:
839:
838:
833:
797:
795:
794:
789:
784:
769:
767:
766:
761:
756:
741:
739:
738:
733:
721:
719:
718:
713:
697:
695:
694:
689:
687:
682:
664:
662:
661:
656:
640:
638:
637:
632:
620:
618:
617:
612:
600:
598:
597:
592:
580:
578:
577:
572:
560:
558:
557:
552:
540:
538:
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532:
520:
518:
517:
512:
494:
492:
491:
486:
474:
472:
471:
466:
420:
418:
417:
412:
400:
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397:
392:
380:
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372:
360:
358:
357:
352:
340:
338:
337:
332:
317:
315:
314:
309:
295:
292:
253:
251:
250:
245:
233:
231:
230:
225:
210:
208:
207:
202:
187:
185:
184:
179:
163:
161:
160:
155:
143:
141:
140:
135:
22:semigroup theory
2004:
2003:
1999:
1998:
1997:
1995:
1994:
1993:
1979:
1978:
1959:
1946:
1943:
1942:
1923:
1922:
1918:
1910:
1893:
1892:
1888:
1882:
1872:Clarendon Press
1862:
1861:
1857:
1841:
1837:
1822:
1798:
1797:
1793:
1762:
1761:
1757:
1752:
1707:is an ideal of
1699:by a semigroup
1689:
1687:Ideal extension
1482:is an ideal of
1080:
1056:
1055:
1036:
1035:
1032:Rees congruence
1012:
1011:
992:
991:
990:The congruence
972:
971:
961:
955:
954:
917:
904:
886:
885:
864:
863:
844:
843:
800:
799:
772:
771:
744:
743:
724:
723:
704:
703:
670:
669:
647:
646:
623:
622:
621:, the relation
603:
602:
601:is an ideal of
583:
582:
563:
562:
543:
542:
523:
522:
497:
496:
477:
476:
457:
456:
403:
402:
383:
382:
381:. The relation
363:
362:
343:
342:
320:
319:
293: and
256:
255:
236:
235:
234:are subsets of
213:
212:
190:
189:
170:
169:
146:
145:
144:of a semigroup
126:
125:
119:
63:be an ideal of
40:, is a certain
36:), named after
12:
11:
5:
2002:
2000:
1992:
1991:
1981:
1980:
1964:
1963:
1957:
1941:
1940:
1916:
1908:
1886:
1880:
1864:Howie, John M.
1855:
1842:Lawson (1998)
1835:
1820:
1791:
1771:(4): 387–400.
1754:
1753:
1751:
1748:
1688:
1685:
1682:
1681:
1675:
1669:
1663:
1657:
1648:
1647:
1641:
1635:
1629:
1623:
1614:
1613:
1607:
1601:
1595:
1589:
1580:
1579:
1573:
1567:
1561:
1555:
1546:
1545:
1540:
1535:
1530:
1525:
1476:
1475:
1418:
1339:
1338:
1332:
1326:
1320:
1314:
1308:
1299:
1298:
1292:
1286:
1280:
1274:
1268:
1259:
1258:
1252:
1246:
1240:
1234:
1228:
1219:
1218:
1212:
1206:
1200:
1194:
1188:
1179:
1178:
1172:
1166:
1160:
1154:
1148:
1139:
1138:
1133:
1128:
1123:
1118:
1113:
1079:
1076:
1063:
1043:
1019:
999:
975:
970:
962:
960:
957:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
918:
916:
913:
910:
909:
907:
902:
899:
896:
893:
871:
851:
831:
828:
825:
822:
819:
816:
813:
810:
807:
787:
783:
779:
759:
755:
751:
731:
711:
686:
681:
677:
654:
630:
610:
590:
570:
550:
530:
510:
507:
504:
484:
464:
453:
452:
410:
390:
370:
350:
330:
327:
307:
304:
301:
298:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
243:
223:
220:
200:
197:
177:
153:
133:
118:
115:
13:
10:
9:
6:
4:
3:
2:
2001:
1990:
1987:
1986:
1984:
1977:
1976:
1974:
1970:
1960:
1954:
1950:
1945:
1944:
1937:
1933:
1932:
1927:
1920:
1917:
1911:
1905:
1901:
1897:
1890:
1887:
1883:
1881:0-19-851194-9
1877:
1873:
1869:
1865:
1859:
1856:
1853:
1849:
1845:
1839:
1836:
1831:
1827:
1823:
1817:
1813:
1809:
1805:
1801:
1795:
1792:
1786:
1782:
1778:
1774:
1770:
1766:
1759:
1756:
1749:
1747:
1745:
1741:
1737:
1733:
1729:
1724:
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1686:
1680:
1676:
1674:
1670:
1668:
1664:
1662:
1658:
1656:
1655:
1650:
1649:
1646:
1642:
1640:
1636:
1634:
1630:
1628:
1624:
1622:
1621:
1616:
1615:
1612:
1608:
1606:
1602:
1600:
1596:
1594:
1590:
1588:
1587:
1582:
1581:
1578:
1574:
1572:
1568:
1566:
1562:
1560:
1556:
1554:
1553:
1548:
1547:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1523:
1522:
1519:
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1419:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1362:
1361:
1360:
1358:
1354:
1350:
1346:
1337:
1333:
1331:
1327:
1325:
1321:
1319:
1315:
1313:
1309:
1307:
1306:
1301:
1300:
1297:
1293:
1291:
1287:
1285:
1281:
1279:
1275:
1273:
1269:
1267:
1266:
1261:
1260:
1257:
1253:
1251:
1247:
1245:
1241:
1239:
1235:
1233:
1229:
1227:
1226:
1221:
1220:
1217:
1213:
1211:
1207:
1205:
1201:
1199:
1195:
1193:
1189:
1187:
1186:
1181:
1180:
1177:
1173:
1171:
1167:
1165:
1161:
1159:
1155:
1153:
1149:
1147:
1146:
1141:
1140:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1111:
1110:
1107:
1105:
1101:
1097:
1093:
1089:
1085:
1077:
1075:
1061:
1041:
1033:
1017:
997:
988:
968:
958:
951:
945:
942:
939:
936:
933:
930:
927:
924:
914:
911:
905:
900:
897:
894:
891:
883:
869:
849:
826:
820:
814:
808:
785:
781:
777:
757:
753:
749:
729:
709:
701:
684:
679:
675:
668:
652:
644:
628:
608:
588:
568:
548:
528:
505:
482:
462:
451:
447:
443:
439:
435:
431:
427:
424:
423:
422:
408:
388:
368:
348:
328:
325:
302:
299:
296:
288:
285:
282:
279:
276:
273:
267:
264:
261:
241:
221:
218:
198:
195:
175:
167:
164:is called an
151:
131:
124:
116:
114:
112:
107:
105:
101:
97:
96:
92:
86:
82:
78:
74:
70:
66:
62:
58:
54:
49:
47:
43:
39:
35:
31:
28:(also called
27:
23:
19:
1966:
1965:
1948:
1929:
1919:
1895:
1889:
1867:
1858:
1843:
1838:
1807:
1794:
1768:
1764:
1758:
1744:cancellation
1725:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1691:A semigroup
1690:
1678:
1672:
1666:
1660:
1653:
1652:
1644:
1638:
1632:
1626:
1619:
1618:
1610:
1604:
1598:
1592:
1585:
1584:
1576:
1570:
1564:
1558:
1551:
1550:
1542:
1537:
1532:
1527:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1477:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1356:
1352:
1348:
1344:
1342:
1335:
1329:
1323:
1317:
1311:
1304:
1303:
1295:
1289:
1283:
1277:
1271:
1264:
1263:
1255:
1249:
1243:
1237:
1231:
1224:
1223:
1215:
1209:
1203:
1197:
1191:
1184:
1183:
1175:
1169:
1163:
1157:
1151:
1144:
1143:
1135:
1130:
1125:
1120:
1115:
1103:
1099:
1095:
1091:
1087:
1083:
1081:
1031:
989:
884:
699:
561:and the set
454:
449:
445:
441:
437:
433:
429:
425:
165:
120:
108:
103:
99:
94:
90:
88:
84:
80:
76:
72:
68:
64:
60:
52:
50:
33:
29:
25:
15:
1846:, page 60,
1494:is the set
421:defined by
83:outside of
34:Rees factor
18:mathematics
1969:PlanetMath
1750:References
1471:} ⊆
1414:} ⊆
643:congruence
111:David Rees
38:David Rees
1936:EMS Press
1789:MR 2, 127
1785:123038112
998:ρ
965:otherwise
949:∖
943:∈
895:∗
870:∗
821:∪
812:∖
758:ρ
685:ρ
629:ρ
483:ρ
389:ρ
300:∈
286:∈
280:∣
113:in 1940.
57:semigroup
42:semigroup
1983:Category
1900:Springer
1866:(1995),
1806:(1961).
1478:the set
1359:. Since
921:if
842:, where
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