1220:
This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.
1516:
of semigroup theory to semiheaps and defined a ρ class to be those elements generating the same principle two-sided ideal. He then proved that no i-simple semiheap can have more than two ρ classes.
1619:
267:
2396:
2056:
1335:
1697:
1965:
1443:
1274:
742:
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409:
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1216:
1106:
979:
377:
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1392:
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2117:
1891:
1871:
933:
913:
765:
429:
2646:
92:
1722:, in 1974 K. A. Zareckii followed Mustafaev's lead to describe ideal equivalence, regularity classes, and ideal factors of a semiheap.
2345:
2574:
1996:
1530:
2584:
2504:
2466:
2682:
2744:
1283:
2235:
1652:
2674:
2497:
1457:. Thus a heap is more than a tweak of a group: it is a general concept including a group as a trivial case.
2749:
2137:
1896:
1409:
1240:
1487:
When the above construction is applied to a heap, the result is in fact a group. Note that the identity
607:
466:
385:
1125:
990:
1158:
1048:
2403:
1711:
1277:
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940:
344:
1744:
1340:
2059:
839:
2708:
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2616:
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1990:
1630:
1513:
807:
48:
770:
434:
8:
2419:
1978:
is required to satisfy only the para-associative law but need not obey the identity law.
321:
298:
28:
2078:
320:, promulgator of semiheaps, heaps, and generalized heaps. Груда contrasts with группа (
2399:
1986:
1876:
1856:
1512:
with an "i-simple semiheap" being one with no proper ideals. Mustafaeva translated the
918:
898:
750:
414:
2654:
2678:
2580:
2500:
2462:
1851:
1498:
1473:
1399:
42:
35:
2722:
2494:
Mathematics across the Iron
Curtain: a history of the algebraic theory of semigroups
1850:
satisfies the identity law but not necessarily the para-associative law, that is, a
324:) which was taken into Russian by transliteration. Indeed, a heap has been called a
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2642:
2063:
1454:
1450:
309:
20:
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2657:
2612:
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2469:
1509:
2459:
2407:
1633:
and the ternary operation of the semiheap applies at the left of a string from
2738:
2692:
1446:
915:
to be the identity of a new group on the set of integers, with the operation
317:
297:
is a semiheap in which every element is biunitary. It can be thought of as a
2669:(1979). "Inverse semigroups and generalised grouds". In A.F. Lavrik (ed.).
2666:
380:
1449:
was motivated to form this heap by his study of transition maps in an
1445:
so a mathematical structure has been formed by the ternary operation.
1981:
An example of a semigroud that is not in general a groud is given by
1505:
32:
1140:
1121:
893:
2342:
A semigroud is a generalised groud if the relation → defined by
1645:
can have at most 5 regularity classes. Mustafaev calls an ideal
308:
is derived from груда, Russian for "heap", "pile", or "stack".
2576:
Mal'cev, Protomodular, Homological and Semi-Abelian
Categories
1120:
The heap of a group may be generalized again to the case of a
2695:(1968). "On the algebraic theory of coordinate atlases, II".
1614:{\displaystyle D(m,n)=\{a\mid \exists x\in S:a=a^{n}xa^{m}\}}
2726:
1491:
of the group can be chosen to be any element of the heap.
1041:
The previous two examples may be generalized to any group
2542:
L. G. Mustafaev (1965) "Regularity classes of semiheaps"
2527:
L. G. Mustafaev (1966) "Ideal equivalences of semiheaps"
262:{\displaystyle \forall a,b,c,d,e\in H\quad ,d,e]=,e]=].}
1725:
1703:= D(2,2), then every ideal is isolated and conversely.
2557:
K. A. Zareckii (1974) "Semiheaps of binary relations"
1508:, the structure of semiheaps is described in terms of
1139:. The elements of the heap may be identified with the
1115:
2645:(1929) "On a generalization of the associative law",
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1999:
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347:
95:
51:
1520:He also described regularity classes of a semiheap
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89:that satisfies a modified associativity property:
81:
2647:Transactions of the American Mathematical Society
1739:satisfies the partial para-associative condition
2736:
2391:{\displaystyle a\rightarrow b\Leftrightarrow =a}
2051:{\displaystyle =x\cdot y^{\mathrm {T} }\cdot z}
2691:
2572:
2514:
2512:
1608:
1555:
1406:. The result of this composition is also in
1330:{\displaystyle p,q,r\in {\mathcal {B}}(A,B)}
366:
354:
16:Algebraic structure with a ternary operation
2598:
2596:
2573:Borceux, Francis; Bourn, Dominique (2004).
2673:. Amer. Math. Soc. Transl. Vol. 113.
2602:
2496:, pages 264,5, History of Mathematics 41,
1692:{\displaystyle a^{n}\in B\implies a\in B.}
1676:
1672:
1224:
892:to produce a heap. We can then choose any
2579:. Springer Science & Business Media.
2509:
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2448:
2446:
2444:
2442:
2440:
2438:
703:
672:
641:
562:
531:
500:
2593:
1108:using the multiplication and inverse of
2603:Moldavs'ka, Z. Ja. "Linear semiheaps".
2454:C.D. Hollings & M.V. Lawson (2017)
1497:: Every semiheap may be embedded in an
1143:from A to B, such that three morphisms
460:. Then it produces the following heap:
301:with the identity element "forgotten".
2737:
2665:
2483:Schein (1979) pp.101–102: footnote (o)
2435:
1468:: A semiheap with a biunitary element
1155:define a heap operation according to
2456:Wagner's Theory of Generalised Heaps
1726:Generalizations and related concepts
1045:by defining the ternary relation as
2406:. In a generalised groud, → is an
1960:{\displaystyle f(x,x,y)=f(y,x,x)=y}
1438:{\displaystyle {\mathcal {B}}(A,B)}
1269:{\displaystyle {\mathcal {B}}(A,B)}
1116:Heap of a groupoid with two objects
799:
336:
13:
2671:Twelve papers in logic and algebra
2036:
1564:
1415:
1307:
1246:
1036:
737:{\displaystyle =b,\,=a,\,=a,\,=b.}
596:{\displaystyle =a,\,=b,\,=b,\,=a,}
391:
96:
14:
2761:
2716:
2655:10.1090/S0002-9947-1929-1501476-0
2486:
2697:Trudy Sem. Vektor. Tenzor. Anal.
2134:is an idempotent semiheap where
796:would have given the same heap.
404:{\displaystyle \mathrm {C} _{2}}
2623:
132:
2566:
2551:
2536:
2521:
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2183:
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2156:
2141:
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2018:
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1903:
1826:
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250:
232:
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169:
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136:
133:
70:
52:
1:
2675:American Mathematical Society
2636:
2498:American Mathematical Society
2492:Christopher Hollings (2014)
1337:define the ternary operator
767:as the identity element and
314:Theory of Generalized Groups
7:
2413:
2323:{\displaystyle ,b,b]=,a,a]}
1460:
1025:{\displaystyle x^{-1}=2k-x}
331:
278:of a semiheap satisfies =
10:
2766:
2605:Dopovidi Ahad. Nauk Ukrain
1211:{\displaystyle =xy^{-1}z.}
1101:{\displaystyle =xy^{-1}z,}
431:the identity element, and
1699:He then proves that when
974:{\displaystyle x*y=x+y-k}
836:are integers, we can set
372:{\displaystyle H=\{a,b\}}
2428:
1893:satisfying the identity
1835:{\displaystyle ,d,e]=].}
1706:Studying the semiheap Z(
1476:with operation given by
1387:{\displaystyle =pq^{T}r}
316:(1937) which influenced
2745:Non-associative algebra
1712:heterogeneous relations
1278:heterogeneous relations
1225:Heterogeneous relations
2392:
2324:
2226:
2113:
2052:
1961:
1887:
1867:
1836:
1693:
1615:
1439:
1388:
1331:
1270:
1237:be different sets and
1212:
1102:
1026:
975:
929:
909:
886:
885:{\displaystyle =x-y+z}
830:
790:
761:
738:
597:
454:
425:
405:
373:
263:
83:
2393:
2325:
2227:
2114:
2060:matrix multiplication
2053:
1962:
1888:
1868:
1837:
1694:
1616:
1480:= and involution by
1472:may be considered an
1440:
1389:
1332:
1271:
1213:
1103:
1027:
976:
930:
910:
887:
831:
829:{\displaystyle x,y,z}
791:
762:
739:
598:
455:
426:
406:
374:
312:used the term in his
264:
84:
82:{\displaystyle \in H}
2346:
2236:
2138:
2079:
2075:is a semiheap where
1997:
1897:
1877:
1857:
1745:
1653:
1531:
1410:
1341:
1284:
1241:
1159:
1049:
991:
941:
919:
899:
840:
808:
789:{\displaystyle aa=b}
771:
751:
608:
467:
453:{\displaystyle bb=a}
435:
415:
386:
345:
93:
49:
2677:. pp. 89–182.
2629:Schein (1979) p.104
2225:{\displaystyle ]=]}
2073:idempotent semiheap
1993:of fixed size with
1504:As in the study of
1499:involuted semigroup
1474:involuted semigroup
29:algebraic structure
2750:Ternary operations
2423:-ary associativity
2402:(idempotence) and
2388:
2320:
2222:
2112:{\displaystyle =a}
2109:
2048:
1957:
1883:
1863:
1832:
1689:
1611:
1435:
1384:
1327:
1280:between them. For
1276:the collection of
1266:
1208:
1098:
1022:
971:
925:
905:
882:
826:
786:
757:
734:
593:
450:
421:
401:
369:
328:in English text.)
259:
79:
2586:978-1-4020-1961-6
2505:978-1-4704-1493-1
2467:978-3-319-63620-7
2132:generalised groud
1886:{\displaystyle X}
1866:{\displaystyle f}
1852:ternary operation
1514:Green's relations
1455:partial functions
1400:converse relation
1135:when viewed as a
928:{\displaystyle *}
908:{\displaystyle k}
760:{\displaystyle b}
424:{\displaystyle a}
273:biunitary element
43:ternary operation
2757:
2712:
2688:
2643:Anton Sushkevich
2630:
2627:
2621:
2620:
2611:: 888–890, 957.
2600:
2591:
2590:
2570:
2564:
2555:
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2534:
2525:
2519:
2516:
2507:
2490:
2484:
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2475:
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2397:
2395:
2394:
2389:
2329:
2327:
2326:
2321:
2231:
2229:
2228:
2223:
2128:generalised heap
2118:
2116:
2115:
2110:
2064:matrix transpose
2058:where • denotes
2057:
2055:
2054:
2049:
2041:
2040:
2039:
1966:
1964:
1963:
1958:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1848:Malcev operation
1841:
1839:
1838:
1833:
1698:
1696:
1695:
1690:
1665:
1664:
1649:"isolated" when
1620:
1618:
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1005:
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911:
906:
891:
889:
888:
883:
835:
833:
832:
827:
800:Heap of integers
795:
793:
792:
787:
766:
764:
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758:
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428:
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337:Two element heap
310:Anton Sushkevich
268:
266:
265:
260:
88:
86:
85:
80:
31:consisting of a
21:abstract algebra
2765:
2764:
2760:
2759:
2758:
2756:
2755:
2754:
2735:
2734:
2723:Mal'cev variety
2719:
2685:
2639:
2634:
2633:
2628:
2624:
2601:
2594:
2587:
2571:
2567:
2556:
2552:
2541:
2537:
2526:
2522:
2517:
2510:
2491:
2487:
2482:
2478:
2453:
2436:
2431:
2416:
2347:
2344:
2343:
2237:
2234:
2233:
2139:
2136:
2135:
2080:
2077:
2076:
2035:
2034:
2030:
1998:
1995:
1994:
1898:
1895:
1894:
1878:
1875:
1874:
1858:
1855:
1854:
1746:
1743:
1742:
1728:
1660:
1656:
1654:
1651:
1650:
1641:He proves that
1602:
1598:
1589:
1585:
1532:
1529:
1528:
1463:
1414:
1413:
1411:
1408:
1407:
1375:
1371:
1342:
1339:
1338:
1306:
1305:
1285:
1282:
1281:
1245:
1244:
1242:
1239:
1238:
1227:
1193:
1189:
1160:
1157:
1156:
1118:
1083:
1079:
1050:
1047:
1046:
1039:
1037:Heap of a group
998:
994:
992:
989:
988:
942:
939:
938:
920:
917:
916:
900:
897:
896:
841:
838:
837:
809:
806:
805:
802:
772:
769:
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752:
749:
748:
609:
606:
605:
468:
465:
464:
436:
433:
432:
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413:
412:
395:
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383:
346:
343:
342:
339:
334:
94:
91:
90:
50:
47:
46:
17:
12:
11:
5:
2763:
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2747:
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2732:
2718:
2717:External links
2715:
2714:
2713:
2699:(in Russian).
2689:
2683:
2663:
2649:31(1): 204–14
2638:
2635:
2632:
2631:
2622:
2607:. RSR Ser. A.
2592:
2585:
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2520:
2508:
2485:
2476:
2460:Springer books
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2408:order relation
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2062:and T denotes
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2011:
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1124:which has two
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640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
603:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
449:
446:
443:
440:
420:
411:, by defining
398:
393:
368:
365:
362:
359:
356:
353:
350:
338:
335:
333:
330:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
98:
78:
75:
72:
69:
66:
63:
60:
57:
54:
15:
9:
6:
4:
3:
2:
2762:
2751:
2748:
2746:
2743:
2742:
2740:
2731:
2729:
2724:
2721:
2720:
2710:
2706:
2702:
2698:
2694:
2693:Wagner, V. V.
2690:
2686:
2684:0-8218-3063-5
2680:
2676:
2672:
2668:
2667:Schein, Boris
2664:
2662:
2659:
2656:
2652:
2648:
2644:
2641:
2640:
2626:
2618:
2614:
2610:
2606:
2599:
2597:
2588:
2582:
2578:
2577:
2569:
2563:
2560:
2554:
2548:
2545:
2539:
2533:
2530:
2524:
2518:Vagner (1968)
2515:
2513:
2506:
2502:
2499:
2495:
2489:
2480:
2474:
2471:
2468:
2464:
2461:
2457:
2451:
2449:
2447:
2445:
2443:
2441:
2439:
2434:
2424:
2422:
2418:
2417:
2411:
2409:
2405:
2404:antisymmetric
2401:
2385:
2382:
2376:
2373:
2370:
2367:
2364:
2355:
2349:
2337:
2333:
2314:
2311:
2308:
2305:
2299:
2296:
2293:
2290:
2287:
2278:
2272:
2269:
2266:
2263:
2257:
2254:
2251:
2248:
2245:
2213:
2210:
2207:
2204:
2201:
2195:
2192:
2189:
2186:
2180:
2171:
2168:
2165:
2162:
2159:
2153:
2150:
2147:
2144:
2133:
2129:
2125:
2122:
2106:
2103:
2097:
2094:
2091:
2088:
2085:
2074:
2070:
2065:
2061:
2045:
2042:
2031:
2027:
2024:
2021:
2015:
2012:
2009:
2006:
2003:
1992:
1988:
1984:
1980:
1979:
1977:
1973:
1969:
1954:
1951:
1945:
1942:
1939:
1936:
1933:
1927:
1924:
1918:
1915:
1912:
1909:
1906:
1900:
1880:
1860:
1853:
1849:
1845:
1829:
1820:
1817:
1814:
1811:
1808:
1802:
1799:
1796:
1793:
1787:
1781:
1778:
1775:
1772:
1766:
1763:
1760:
1757:
1754:
1741:
1740:
1738:
1734:
1730:
1729:
1723:
1721:
1717:
1714:between sets
1713:
1709:
1704:
1702:
1686:
1683:
1680:
1677:
1669:
1666:
1661:
1657:
1648:
1644:
1636:
1632:
1628:
1624:
1603:
1599:
1595:
1590:
1586:
1582:
1579:
1576:
1573:
1570:
1567:
1561:
1558:
1552:
1546:
1543:
1540:
1534:
1527:
1526:
1525:
1523:
1518:
1515:
1511:
1507:
1502:
1500:
1496:
1492:
1490:
1485:
1483:
1479:
1475:
1471:
1467:
1458:
1456:
1452:
1448:
1447:Viktor Wagner
1429:
1426:
1423:
1405:
1401:
1397:
1381:
1376:
1372:
1368:
1365:
1359:
1356:
1353:
1350:
1347:
1321:
1318:
1315:
1302:
1299:
1296:
1293:
1290:
1287:
1279:
1260:
1257:
1254:
1236:
1232:
1222:
1218:
1205:
1202:
1197:
1194:
1190:
1186:
1183:
1177:
1174:
1171:
1168:
1165:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1127:
1123:
1113:
1111:
1095:
1092:
1087:
1084:
1080:
1076:
1073:
1067:
1064:
1061:
1058:
1055:
1044:
1019:
1016:
1013:
1010:
1007:
1002:
999:
995:
987:
986:
985:
968:
965:
962:
959:
956:
953:
950:
947:
944:
937:
936:
935:
922:
902:
895:
879:
876:
873:
870:
867:
864:
858:
855:
852:
849:
846:
823:
820:
817:
814:
811:
797:
783:
780:
777:
774:
754:
731:
728:
725:
719:
716:
713:
710:
707:
700:
697:
694:
688:
685:
682:
679:
676:
669:
666:
663:
657:
654:
651:
648:
645:
638:
635:
632:
626:
623:
620:
617:
614:
604:
590:
587:
584:
578:
575:
572:
569:
566:
559:
556:
553:
547:
544:
541:
538:
535:
528:
525:
522:
516:
513:
510:
507:
504:
497:
494:
491:
485:
482:
479:
476:
473:
463:
462:
461:
447:
444:
441:
438:
418:
396:
382:
363:
360:
357:
351:
348:
329:
327:
323:
319:
318:Viktor Wagner
315:
311:
307:
302:
300:
296:
291:
289:
285:
282:= for every
281:
277:
274:
269:
256:
247:
244:
241:
238:
235:
229:
226:
223:
220:
214:
208:
205:
199:
196:
193:
190:
187:
181:
178:
172:
166:
163:
160:
157:
151:
148:
145:
142:
139:
129:
126:
123:
120:
117:
114:
111:
108:
105:
102:
99:
76:
73:
67:
64:
61:
58:
55:
44:
40:
37:
34:
30:
26:
22:
2727:
2700:
2696:
2670:
2625:
2608:
2604:
2575:
2568:
2553:
2538:
2523:
2493:
2488:
2479:
2455:
2420:
2341:
2335:
2331:
2131:
2127:
2120:
2072:
1982:
1975:
1971:
1847:
1736:
1732:
1719:
1715:
1707:
1705:
1700:
1646:
1642:
1640:
1634:
1626:
1622:
1521:
1519:
1503:
1494:
1493:
1488:
1486:
1481:
1477:
1469:
1465:
1464:
1403:
1395:
1234:
1230:
1228:
1219:
1152:
1148:
1144:
1132:
1128:
1119:
1109:
1042:
1040:
984:and inverse
983:
803:
746:
381:cyclic group
340:
325:
313:
305:
303:
294:
292:
287:
283:
279:
275:
272:
270:
38:
24:
18:
2703:: 229–281.
1737:pseudogroud
2739:Categories
2637:References
1733:pseudoheap
1506:semigroups
1453:which are
2400:reflexive
2359:⇔
2353:→
2043:⋅
2028:⋅
1976:semigroud
1873:on a set
1681:∈
1674:⟹
1667:∈
1571:∈
1565:∃
1562:∣
1303:∈
1195:−
1141:morphisms
1085:−
1017:−
1000:−
966:−
948:∗
923:∗
871:−
747:Defining
379:into the
304:The term
127:∈
97:∀
74:∈
33:non-empty
2414:See also
2330:for all
2119:for all
1991:matrices
1972:semiheap
1461:Theorems
1137:category
1122:groupoid
332:Examples
45:denoted
25:semiheap
2725:at the
2709:0253970
2661:1501476
2617:0297918
2562:0364526
2547:0209386
2532:0202892
2473:3729305
1495:Theorem
1466:Theorem
1398:is the
1126:objects
894:integer
41:with a
2707:
2681:
2615:
2583:
2503:
2465:
1631:parity
1621:where
1510:ideals
1394:where
27:is an
2429:Notes
1710:) of
1451:atlas
341:Turn
326:groud
322:group
299:group
2679:ISBN
2609:1971
2581:ISBN
2501:ISBN
2463:ISBN
2334:and
2232:and
1987:ring
1718:and
1708:A, B
1625:and
1484:= .
1233:and
1229:Let
1131:and
306:heap
295:heap
23:, a
2730:Lab
2651:doi
2398:is
2130:or
2071:An
1989:of
1974:or
1735:or
1402:of
804:If
286:in
36:set
19:In
2741::
2705:MR
2701:14
2658:MR
2613:MR
2595:^
2559:MR
2544:MR
2529:MR
2511:^
2470:MR
2458:,
2437:^
2410:.
2126:A
1985:a
1970:A
1846:A
1731:A
1524::
1501:.
1478:ab
1151:,
1147:,
1112:.
293:A
290:.
271:A
2728:n
2711:.
2687:.
2653::
2619:.
2589:.
2421:n
2386:a
2383:=
2380:]
2377:a
2374:,
2371:b
2368:,
2365:a
2362:[
2356:b
2350:a
2338:.
2336:b
2332:a
2318:]
2315:a
2312:,
2309:a
2306:,
2303:]
2300:b
2297:,
2294:b
2291:,
2288:x
2285:[
2282:[
2279:=
2276:]
2273:b
2270:,
2267:b
2264:,
2261:]
2258:a
2255:,
2252:a
2249:,
2246:x
2243:[
2240:[
2220:]
2217:]
2214:x
2211:,
2208:a
2205:,
2202:a
2199:[
2196:,
2193:b
2190:,
2187:b
2184:[
2181:=
2178:]
2175:]
2172:x
2169:,
2166:b
2163:,
2160:b
2157:[
2154:,
2151:a
2148:,
2145:a
2142:[
2123:.
2121:a
2107:a
2104:=
2101:]
2098:a
2095:,
2092:a
2089:,
2086:a
2083:[
2066:.
2046:z
2037:T
2032:y
2025:x
2022:=
2019:]
2016:z
2013:,
2010:y
2007:,
2004:x
2001:[
1983:M
1967:.
1955:y
1952:=
1949:)
1946:x
1943:,
1940:x
1937:,
1934:y
1931:(
1928:f
1925:=
1922:)
1919:y
1916:,
1913:x
1910:,
1907:x
1904:(
1901:f
1881:X
1861:f
1830:.
1827:]
1824:]
1821:e
1818:,
1815:d
1812:,
1809:c
1806:[
1803:,
1800:b
1797:,
1794:a
1791:[
1788:=
1785:]
1782:e
1779:,
1776:d
1773:,
1770:]
1767:c
1764:,
1761:b
1758:,
1755:a
1752:[
1749:[
1720:B
1716:A
1701:S
1687:.
1684:B
1678:a
1670:B
1662:n
1658:a
1647:B
1643:S
1637:.
1635:S
1627:m
1623:n
1609:}
1604:m
1600:a
1596:x
1591:n
1587:a
1583:=
1580:a
1577::
1574:S
1568:x
1559:a
1556:{
1553:=
1550:)
1547:n
1544:,
1541:m
1538:(
1535:D
1522:S
1489:e
1482:a
1470:e
1433:)
1430:B
1427:,
1424:A
1421:(
1416:B
1404:q
1396:q
1382:r
1377:T
1373:q
1369:p
1366:=
1363:]
1360:r
1357:,
1354:q
1351:,
1348:p
1345:[
1325:)
1322:B
1319:,
1316:A
1313:(
1308:B
1300:r
1297:,
1294:q
1291:,
1288:p
1264:)
1261:B
1258:,
1255:A
1252:(
1247:B
1235:B
1231:A
1206:.
1203:z
1198:1
1191:y
1187:x
1184:=
1181:]
1178:z
1175:,
1172:y
1169:,
1166:x
1163:[
1153:z
1149:y
1145:x
1133:B
1129:A
1110:G
1096:,
1093:z
1088:1
1081:y
1077:x
1074:=
1071:]
1068:z
1065:,
1062:y
1059:,
1056:x
1053:[
1043:G
1032:.
1020:x
1014:k
1011:2
1008:=
1003:1
996:x
969:k
963:y
960:+
957:x
954:=
951:y
945:x
903:k
880:z
877:+
874:y
868:x
865:=
862:]
859:z
856:,
853:y
850:,
847:x
844:[
824:z
821:,
818:y
815:,
812:x
784:b
781:=
778:a
775:a
755:b
732:.
729:b
726:=
723:]
720:b
717:,
714:b
711:,
708:b
705:[
701:,
698:a
695:=
692:]
689:a
686:,
683:b
680:,
677:b
674:[
670:,
667:a
664:=
661:]
658:b
655:,
652:b
649:,
646:a
643:[
639:,
636:b
633:=
630:]
627:a
624:,
621:b
618:,
615:a
612:[
591:,
588:a
585:=
582:]
579:b
576:,
573:a
570:,
567:b
564:[
560:,
557:b
554:=
551:]
548:a
545:,
542:a
539:,
536:b
533:[
529:,
526:b
523:=
520:]
517:b
514:,
511:a
508:,
505:a
502:[
498:,
495:a
492:=
489:]
486:a
483:,
480:a
477:,
474:a
471:[
448:a
445:=
442:b
439:b
419:a
397:2
392:C
367:}
364:b
361:,
358:a
355:{
352:=
349:H
288:H
284:k
280:k
276:h
257:.
254:]
251:]
248:e
245:,
242:d
239:,
236:c
233:[
230:,
227:b
224:,
221:a
218:[
215:=
212:]
209:e
206:,
203:]
200:b
197:,
194:c
191:,
188:d
185:[
182:,
179:a
176:[
173:=
170:]
167:e
164:,
161:d
158:,
155:]
152:c
149:,
146:b
143:,
140:a
137:[
134:[
130:H
124:e
121:,
118:d
115:,
112:c
109:,
106:b
103:,
100:a
77:H
71:]
68:z
65:,
62:y
59:,
56:x
53:[
39:H
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