1196:. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. p. 459.
110:
1143:
under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.
61:
1135:
The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. Where the definition of
84:
There are at least three standard ways of constructing the bicyclic semigroup, and various notations for referring to it. Lyapin called it
533:
uses two appropriately-chosen functions to yield the bicyclic semigroup as a monoid of transformations of the natural numbers. Let
1201:
1139:
from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative
1124:
appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each
1157:
428:
It is not entirely obvious that this should be the case – perhaps the hardest task is understanding that
150:. Thus, each semigroup element is a string of those two letters, with the proviso that the subsequence "
73:
1293:
1240:, A. H. Clifford and G. B. Preston. American Mathematical Society, 1961 (volume 1), 1967 (volume 2).
91:
546:
529:
Note that the two definitions given above both satisfy these properties. A third way of deriving
709:). In practice, this means that the bicyclic semigroup can be found in many different contexts.
1030:
are infinitely many copies of the trivial group, each corresponding to one of the idempotents.
32:, it is usually referred to as simply a semigroup. It is perhaps most easily understood as the
824:
1152:
1034:
917:
1211:
159:" does not appear. The semigroup operation is concatenation of strings, which is clearly
8:
137:
49:
1272:
65:
599:
These three functions have the required properties, so the semigroup they generate is
1197:
773:
1207:
627:
313:
originally, with the empty string as the monoid identity, this new construction of
33:
1140:
1026:-related if and only if they are identical. Consequently, the only subgroups of
1250:
Canonical form of elements of an associative system given by defining relations
973:
177:
76:, discovered it independently (without publication) at some point before 1943.
69:
241:
is the semigroup of pairs of natural numbers (including zero), with operation
1287:
1263:
Hollings, C.D. (2007). "Some First
Tantalizing Steps into Semigroup Theory".
41:
37:
612:
1189:
358:
160:
125:
45:
17:
1276:
40:
of balanced pairs of parentheses. Thus, it finds common applications in
713:
522:
which is not allowed – so there are infinitely many distinct powers of
301:
so that it is the same object as in the original construction. Just as
25:
221:
The way in which these exponents are constrained suggests that the "
736:
is any natural number (using the ordered pair characterisation of
29:
611:
The bicyclic semigroup has the property that the image of any
229:
structure" can be discarded, leaving only operations on the "
24:
is an algebraic object important for the structure theory of
60:
The first published description of this object was given by
526:. The full proof is given in Clifford and Preston's book.
1170:
831:
is principal: the left and right principal ideals of
701:; otherwise, it is the cyclic semigroup generated by
116:. This article will use the modern style throughout.
94:
1246:, Pierre Antoine Grillet. Marcel Dekker, Inc., 1995.
666:
is a homomorphism) with the possible exception that
1244:
Semigroups: an introduction to the structure theory
1041:is infinitely large; the upper left corner begins:
909:Each of these contains infinitely many others, so
662:will always satisfy the conditions above (because
104:
1285:
1271:. Mathematical Association of America: 331–344.
124:The bicyclic semigroup is the quotient of the
163:. It can then be shown that all elements of
1132:-class must contain exactly one idempotent.
913:does not have minimal left or right ideals.
432:must be infinite. To see this, suppose that
112:; and most recent papers have tended to use
187:. The composition operation simplifies to
1262:
1188:
1176:
119:
436:(say) does not have infinite order, so
357:that satisfies the statements below is
1286:
1182:
788:, in the "weak" semigroup sense that
216:
72:claim that one of them, working with
1254:Leningrad Gos. Ped. Inst. Uch. Zap.
1022:This implies that two elements are
13:
1238:The algebraic theory of semigroups
1108:Each entry represents a singleton
97:
14:
1305:
630:, or it is an isomorphic copy of
332:
1194:Algebraic combinatorics on words
1045:
776:. (This means that each element
772:), the bicyclic semigroup is an
79:
1218:
549:on the natural numbers, where
105:{\displaystyle {\mathcal {C}}}
1:
1252:, Evgenii Sergeevich Lyapin,
1231:
1158:Special classes of semigroups
1120:-classes. The idempotents of
1116:-classes and the columns are
689:). If this is not true, then
606:
297:This is sufficient to define
740:). Since these commute, and
88:; Clifford and Preston used
7:
1146:
361:to the bicyclic semigroup.
28:. Although it is in fact a
10:
1310:
932:), and hence has only one
140:generated by the relation
55:
1112:-class; the rows are the
1163:
547:transformation semigroup
948:relations are given by
685:might turn out to be φ(
345:generated by elements
167:in fact have the form
106:
1259:(1953), pages 45–54 .
1153:Four-spiral semigroup
784:has a unique inverse
622:to another semigroup
337:It can be shown that
120:From a free semigroup
107:
44:, such as describing
1265:Mathematics Magazine
595:− 1 otherwise.
92:
50:associative algebras
697:) is isomorphic to
545:be elements of the
217:From ordered pairs
128:on two generators
102:
66:Alfred H. Clifford
22:bicyclic semigroup
1203:978-0-521-18071-9
1104:
1103:
1008:) if and only if
918:Green's relations
774:inverse semigroup
1301:
1294:Semigroup theory
1280:
1225:
1222:
1216:
1215:
1186:
1180:
1174:
1128:-class and each
1046:
1017:
984:
842:
819:
803:
771:
731:
684:
463:
445:
328:
325:, with identity
324:
320:
175:
158:
149:
111:
109:
108:
103:
101:
100:
34:syntactic monoid
1309:
1308:
1304:
1303:
1302:
1300:
1299:
1298:
1284:
1283:
1234:
1229:
1228:
1223:
1219:
1204:
1187:
1183:
1177:Hollings (2007)
1175:
1171:
1166:
1149:
1035:egg-box diagram
1009:
976:
832:
805:
789:
757:
721:
667:
634:. The elements
609:
455:
437:
335:
326:
322:
318:
317:has generators
219:
178:natural numbers
168:
151:
141:
122:
96:
95:
93:
90:
89:
82:
58:
36:describing the
12:
11:
5:
1307:
1297:
1296:
1282:
1281:
1260:
1247:
1241:
1233:
1230:
1227:
1226:
1217:
1202:
1181:
1168:
1167:
1165:
1162:
1161:
1160:
1155:
1148:
1145:
1106:
1105:
1102:
1101:
1098:
1095:
1092:
1088:
1087:
1084:
1081:
1078:
1074:
1073:
1070:
1067:
1064:
1060:
1059:
1056:
1053:
1050:
1020:
1019:
986:
974:if and only if
936:-class (it is
928:-class (it is
907:
906:
876:
720:are all pairs
608:
605:
597:
596:
578:
564:
520:
519:
498:
497:
426:
425:
413:
401:
382:
334:
333:From functions
331:
295:
294:
218:
215:
214:
213:
121:
118:
99:
81:
78:
70:Gordon Preston
62:Evgenii Lyapin
57:
54:
9:
6:
4:
3:
2:
1306:
1295:
1292:
1291:
1289:
1278:
1274:
1270:
1266:
1261:
1258:
1255:
1251:
1248:
1245:
1242:
1239:
1236:
1235:
1221:
1213:
1209:
1205:
1199:
1195:
1191:
1185:
1179:, p. 332
1178:
1173:
1169:
1159:
1156:
1154:
1151:
1150:
1144:
1142:
1138:
1133:
1131:
1127:
1123:
1119:
1115:
1111:
1099:
1096:
1093:
1090:
1089:
1085:
1082:
1079:
1076:
1075:
1071:
1068:
1065:
1062:
1061:
1057:
1054:
1051:
1048:
1047:
1044:
1043:
1042:
1040:
1036:
1031:
1029:
1025:
1016:
1012:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
951:
950:
949:
947:
943:
939:
935:
931:
927:
924:has only one
923:
919:
914:
912:
904:
900:
896:
892:
888:
884:
880:
877:
874:
870:
866:
862:
858:
854:
850:
846:
845:
844:
840:
836:
830:
826:
821:
818:
814:
811:
808:
802:
798:
795:
792:
787:
783:
779:
775:
770:
766:
763:
760:
755:
751:
747:
743:
739:
735:
729:
725:
719:
715:
710:
708:
704:
700:
696:
692:
688:
682:
678:
674:
670:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
625:
621:
617:
614:
604:
602:
594:
590:
586:
582:
579:
576:
572:
568:
565:
563:
559:
555:
552:
551:
550:
548:
544:
540:
536:
532:
527:
525:
517:
513:
509:
506:
503:
502:
501:
495:
491:
488:
484:
481:
477:
474:
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467:
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458:
453:
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414:
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405:
402:
400:
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360:
356:
352:
348:
344:
340:
330:
316:
312:
308:
304:
300:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
243:
242:
240:
236:
232:
228:
224:
211:
208:
204:
201:
197:
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190:
189:
188:
186:
182:
179:
174:
171:
166:
162:
157:
154:
147:
144:
139:
135:
131:
127:
117:
115:
87:
77:
75:
71:
67:
63:
53:
51:
47:
43:
42:combinatorics
39:
38:Dyck language
35:
31:
27:
23:
19:
1268:
1264:
1256:
1253:
1249:
1243:
1237:
1220:
1193:
1190:Lothaire, M.
1184:
1172:
1136:
1134:
1129:
1125:
1121:
1117:
1113:
1109:
1107:
1038:
1032:
1027:
1023:
1021:
1014:
1010:
1005:
1001:
997:
993:
989:
981:
977:
969:
965:
961:
957:
953:
945:
941:
937:
933:
929:
925:
921:
916:In terms of
915:
910:
908:
902:
898:
894:
890:
886:
882:
878:
872:
868:
864:
860:
856:
852:
848:
838:
834:
828:
822:
816:
812:
809:
806:
800:
796:
793:
790:
785:
781:
777:
768:
764:
761:
758:
753:
749:
745:
741:
737:
733:
727:
723:
717:
711:
706:
702:
698:
694:
690:
686:
680:
676:
672:
668:
663:
659:
655:
651:
647:
643:
639:
635:
631:
623:
619:
615:
613:homomorphism
610:
600:
598:
592:
588:
584:
580:
574:
570:
566:
561:
557:
553:
542:
538:
534:
530:
528:
523:
521:
515:
511:
507:
504:
499:
493:
489:
486:
482:
479:
475:
472:
468:
460:
456:
451:
447:
442:
438:
433:
429:
427:
422:
418:
415:
410:
406:
403:
398:
394:
391:
387:
384:
379:
375:
372:
368:
365:
354:
350:
346:
342:
338:
336:
314:
310:
306:
302:
298:
296:
290:
286:
285:− min{
282:
278:
274:
270:
269:− min{
266:
262:
258:
254:
250:
246:
238:
237:" part. So
234:
230:
226:
222:
220:
209:
206:
202:
199:
195:
192:
184:
180:
172:
169:
164:
155:
152:
145:
142:
133:
129:
123:
113:
85:
83:
80:Construction
59:
46:binary trees
21:
15:
1224:Howie p. 60
752:there is a
748:(for every
714:idempotents
176:, for some
161:associative
126:free monoid
18:mathematics
1232:References
1212:1221.68183
756:such that
626:is either
607:Properties
359:isomorphic
341:semigroup
309:generated
138:congruence
74:David Rees
26:semigroups
897:) :
867:) :
591:= 0, and
587:) = 0 if
446:for some
64:in 1953.
1288:Category
1277:27643058
1192:(2011).
1147:See also
930:bisimple
732:, where
1141:lattice
940:). The
746:regular
454:. Then
136:by the
56:History
1275:
1210:
1200:
1083:(2, 2)
1080:(1, 2)
1077:(0, 2)
1069:(2, 1)
1066:(1, 1)
1063:(0, 1)
1055:(2, 0)
1052:(1, 0)
1049:(0, 0)
938:simple
889:) = {(
823:Every
650:) and
628:cyclic
541:, and
464:, and
353:, and
327:(0, 0)
323:(0, 1)
319:(1, 0)
30:monoid
20:, the
1273:JSTOR
1164:Notes
985:; and
875:} and
825:ideal
658:) of
618:from
261:) = (
1198:ISBN
1100:...
1086:...
1072:...
1058:...
1037:for
1033:The
944:and
859:= {(
843:are
804:and
712:The
573:) =
560:) =
500:so
450:and
321:and
305:and
233:and
225:and
205:) =
183:and
132:and
68:and
48:and
1208:Zbl
1097:...
1094:...
1091:...
827:of
820:.)
780:of
744:is
716:of
642:),
577:+ 1
339:any
293:}).
277:},
253:) (
198:) (
148:= 1
16:In
1290::
1269:80
1267:.
1257:89
1206:.
1013:=
1004:,
996:)
992:,
980:=
972:)
968:,
960:)
956:,
920:,
905:}.
901:≥
893:,
885:,
871:≥
863:,
855:)
851:,
837:,
815:=
799:=
767:=
726:,
675:)
603:.
537:,
514:=
510:=
492:=
485:=
478:=
471:=
459:=
441:=
421:≠
409:=
397:=
390:=
378:=
371:=
349:,
329:.
289:,
281:+
273:,
265:+
257:,
249:,
52:.
1279:.
1214:.
1137:B
1130:R
1126:L
1122:B
1118:L
1114:R
1110:H
1039:B
1028:B
1024:H
1018:.
1015:d
1011:b
1006:d
1002:c
1000:(
998:L
994:b
990:a
988:(
982:c
978:a
970:d
966:c
964:(
962:R
958:b
954:a
952:(
946:R
942:L
934:J
926:D
922:B
911:B
903:n
899:t
895:t
891:s
887:n
883:m
881:(
879:B
873:m
869:s
865:t
861:s
857:B
853:n
849:m
847:(
841:)
839:n
835:m
833:(
829:B
817:y
813:y
810:x
807:y
801:x
797:x
794:y
791:x
786:y
782:B
778:x
769:x
765:x
762:y
759:x
754:y
750:x
742:B
738:B
734:x
730:)
728:x
724:x
722:(
718:B
707:a
705:(
703:φ
699:B
695:B
693:(
691:φ
687:e
683:)
681:a
679:(
677:φ
673:b
671:(
669:φ
664:φ
660:S
656:e
654:(
652:φ
648:b
646:(
644:φ
640:a
638:(
636:φ
632:B
624:S
620:B
616:φ
601:B
593:n
589:n
585:n
583:(
581:β
575:n
571:n
569:(
567:α
562:n
558:n
556:(
554:ι
543:ι
539:β
535:α
531:B
524:a
518:,
516:e
512:a
508:a
505:b
496:,
494:a
490:e
487:a
483:b
480:a
476:b
473:e
469:b
461:e
457:a
452:k
448:h
443:a
439:a
434:a
430:S
423:e
419:a
416:b
411:e
407:b
404:a
399:b
395:b
392:e
388:e
385:b
380:a
376:a
373:e
369:e
366:a
355:b
351:a
347:e
343:S
315:B
311:B
307:q
303:p
299:B
291:c
287:b
283:b
279:d
275:c
271:b
267:c
263:a
259:d
255:c
251:b
247:a
245:(
239:B
235:b
231:a
227:q
223:p
212:.
210:p
207:q
203:p
200:q
196:p
193:q
191:(
185:b
181:a
173:p
170:q
165:B
156:q
153:p
146:q
143:p
134:q
130:p
114:B
98:C
86:P
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