2774:
for varieties having congruence lattices that are distributive (thus called congruence-distributive varieties), while in 1969 Alan Day did the same for varieties having congruence lattices that are modular. Generically, such conditions are called
Maltsev conditions.
2041:
2359:
2234:
1494:
382:
2440:
1262:
1133:
970:
2526:
783:
1729:
1623:
66:
partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.
2612:
2650:
1945:
2263:
1675:
2107:
1912:
1794:
2150:
2727:; this is called a Maltsev term and varieties with this property are called Maltsev varieties. Maltsev's characterization explains a large number of similar results in groups (take
563:
216:
429:
1358:
2558:
2467:
2258:
2145:
1936:
1830:
1753:
1577:
1549:
1521:
1326:
1294:
1161:
691:
641:
617:
517:
135:
461:
248:
1017:
1864:
1043:
667:
1371:
593:
2766:
After
Maltsev's result, other researchers found characterizations based on conditions similar to that found by Maltsev but for other kinds of properties. In 1967
2669:
established the following characterization of congruence-permutable varieties: a variety is congruence permutable if and only if there exist a ternary term
253:
2364:
1166:
2984:
2965:
2944:
2913:
1063:
788:
2472:
696:
1688:
1582:
463:. An equivalence relation compatible with all the operations of an algebra is called a congruence with respect to this algebra.
86:
2579:
2036:{\displaystyle \wedge :\mathrm {Con} ({\mathcal {A}})\times \mathrm {Con} ({\mathcal {A}})\to \mathrm {Con} ({\mathcal {A}})}
2617:
2354:{\displaystyle \vee :\mathrm {Con} ({\mathcal {A}})\times \mathrm {Con} ({\mathcal {A}})\to \mathrm {Con} ({\mathcal {A}})}
1640:
2820:
A. G. Kurosh, Lectures on
General Algebra, Translated from the Russian edition (Moscow, 1960), Chelsea, New York, 1963.
2046:
2659:
1876:
3003:
2796:
1765:
2229:{\displaystyle \langle E\rangle _{\mathcal {A}}=\bigcap \{F\in \mathrm {Con} ({\mathcal {A}})\mid E\subseteq F\}}
2759:
etc. Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is
2573:
2779:
59:
534:
168:
2654:
387:
98:
1331:
2801:
2752:
1759:
51:
2539:
2448:
2239:
2126:
1917:
1811:
1734:
1558:
1530:
1502:
1307:
1275:
1142:
672:
622:
598:
498:
116:
2236:. Note that the closure of a binary relation is a congruence and thus depends on the operations in
1634:
1552:
434:
221:
43:
39:
2886:
1939:
975:
35:
1843:
2980:
2961:
2955:
2940:
2909:
2767:
1837:
1489:{\displaystyle \mathop {\mathrm {ker} } \,h=\{(a,a')\in A^{2}\,|\,h(a)=h(a')\}\subseteq A^{2}}
480:
63:
2934:
2876:
2843:
2662:
is said to be congruence-permutable if all its members are congruence-permutable algebras.
2113:
1365:
1022:
646:
69:
The idea of the quotient algebra abstracts into one common notion the quotient structure of
571:
2760:
2756:
2666:
2529:
2117:
94:
2771:
16:
Result of partitioning the elements of an algebraic structure using a congruence relation
90:
78:
2112:
On the other hand, congruences are not closed under union. However, we can define the
2997:
2890:
2791:
70:
20:
1301:
484:
82:
377:{\displaystyle (f(a_{1},a_{2},\ldots ,a_{n}),f(b_{1},b_{2},\ldots ,b_{n}))\in E}
74:
27:
2848:
2741:
2435:{\displaystyle E_{1}\vee E_{2}=\langle E_{1}\cup E_{2}\rangle _{\mathcal {A}}}
1678:
1626:
2831:
2614:, then their join (in the congruence lattice) is equal to their composition:
1360:
mapping every element to its equivalence class. In fact, every homomorphism
519:, it is straightforward to define the operations induced on the elements of
2881:
1257:{\displaystyle {\mathcal {A}}/E=(A/E,(f_{i}^{{\mathcal {A}}/E})_{i\in I})}
2864:
2782:
for generating
Maltsev conditions associated with congruence identities.
1866:
are trivial congruences. An algebra with no other congruences is called
2865:"A Characterization of Modularity for Congruence Lattices of Algebras"
1938:. Because congruences are closed under intersection, we can define a
1128:{\displaystyle {\mathcal {A}}=(A,(f_{i}^{\mathcal {A}})_{i\in I})}
566:
619:(where the superscript simply denotes that it is an operation in
965:{\displaystyle f_{i}^{{\mathcal {A}}/E}(_{E},\ldots ,_{E})=_{E}}
2521:{\displaystyle (\mathrm {Con} ({\mathcal {A}}),\wedge ,\vee )}
778:{\displaystyle f_{i}^{{\mathcal {A}}/E}:(A/E)^{n_{i}}\to A/E}
483:. The set of these equivalence classes is usually called the
2545:
2495:
2454:
2426:
2343:
2316:
2289:
2245:
2203:
2166:
2132:
2025:
1998:
1971:
1923:
1896:
1817:
1740:
1694:
1662:
1652:
1588:
1564:
1536:
1508:
1337:
1313:
1281:
1220:
1172:
1148:
1100:
1069:
904:
804:
712:
678:
628:
604:
549:
504:
122:
1724:{\displaystyle {\mathcal {A}}/\mathop {\mathrm {ker} } \,h}
1618:{\displaystyle {\mathcal {A}}/\mathop {\mathrm {ker} } \,h}
62:
of the algebra, in the formal sense described below. Its
2979:. CRC Press. pp. 122–124, 137 (Maltsev varieties).
2607:{\displaystyle \alpha \circ \beta =\beta \circ \alpha }
2658:
if every pair of its congruences permutes; likewise a
2645:{\displaystyle \alpha \circ \beta =\alpha \vee \beta }
2147:, such that it is a congruence, in the following way:
1681:
homomorphism. Then, there exists a unique isomorphism
2832:"Algebras Whose Congruence Lattices are Distributive"
2620:
2582:
2542:
2475:
2451:
2367:
2266:
2242:
2153:
2129:
2049:
2043:
by simply taking the intersection of the congruences
1948:
1920:
1879:
1846:
1814:
1768:
1737:
1691:
1643:
1585:
1561:
1533:
1505:
1374:
1334:
1310:
1278:
1169:
1145:
1066:
1025:
978:
791:
699:
675:
649:
625:
601:
574:
537:
501:
437:
390:
256:
224:
171:
119:
2977:
Universal
Algebra: Fundamentals and Selected Topics
19:For quotient associative algebras over a ring, see
2644:
2606:
2552:
2520:
2461:
2434:
2353:
2252:
2228:
2139:
2101:
2035:
1930:
1906:
1858:
1824:
1788:
1747:
1723:
1670:{\displaystyle h:{\mathcal {A}}\to {\mathcal {B}}}
1669:
1617:
1571:
1543:
1515:
1488:
1352:
1320:
1288:
1256:
1155:
1127:
1037:
1011:
964:
777:
685:
661:
635:
611:
587:
557:
511:
455:
423:
376:
242:
210:
129:
2102:{\displaystyle E_{1}\wedge E_{2}=E_{1}\cap E_{2}}
531:is a congruence. Specifically, for any operation
2528:with the two operations defined above forms a
1907:{\displaystyle \mathrm {Con} ({\mathcal {A}})}
8:
2421:
2394:
2223:
2178:
2161:
2154:
1789:{\displaystyle \mathop {\mathrm {ker} } \,h}
1470:
1395:
2763:as well; the converse is not true however.
50:. Here, the congruence relation must be an
2260:, not just on the carrier set. Now define
404:
188:
2933:Klaus Denecke; Shelly L. Wismath (2009).
2880:
2847:
2619:
2581:
2544:
2543:
2541:
2494:
2493:
2479:
2474:
2453:
2452:
2450:
2425:
2424:
2414:
2401:
2385:
2372:
2366:
2342:
2341:
2327:
2315:
2314:
2300:
2288:
2287:
2273:
2265:
2244:
2243:
2241:
2202:
2201:
2187:
2165:
2164:
2152:
2131:
2130:
2128:
2093:
2080:
2067:
2054:
2048:
2024:
2023:
2009:
1997:
1996:
1982:
1970:
1969:
1955:
1947:
1922:
1921:
1919:
1914:be the set of congruences on the algebra
1895:
1894:
1880:
1878:
1845:
1816:
1815:
1813:
1782:
1770:
1769:
1767:
1762:with the natural homomorphism induced by
1739:
1738:
1736:
1717:
1705:
1704:
1699:
1693:
1692:
1690:
1661:
1660:
1651:
1650:
1642:
1611:
1599:
1598:
1593:
1587:
1586:
1584:
1563:
1562:
1560:
1535:
1534:
1532:
1527:thus defines two algebras homomorphic to
1507:
1506:
1504:
1480:
1437:
1432:
1431:
1425:
1388:
1376:
1375:
1373:
1364:determines a congruence relation via the
1342:
1336:
1335:
1333:
1312:
1311:
1309:
1280:
1279:
1277:
1239:
1225:
1219:
1218:
1217:
1212:
1194:
1177:
1171:
1170:
1168:
1147:
1146:
1144:
1110:
1099:
1098:
1093:
1068:
1067:
1065:
1024:
1001:
989:
977:
956:
941:
936:
917:
903:
902:
897:
878:
866:
861:
839:
829:
809:
803:
802:
801:
796:
790:
767:
753:
748:
736:
717:
711:
710:
709:
704:
698:
677:
676:
674:
648:
627:
626:
624:
603:
602:
600:
579:
573:
548:
547:
542:
536:
503:
502:
500:
436:
409:
395:
389:
356:
337:
324:
302:
283:
270:
255:
223:
193:
179:
170:
121:
120:
118:
113:be the set of the elements of an algebra
2908:. American Mathematical Soc. p. 4.
2813:
2960:. PHI Learning Pvt. Ltd. p. 215.
141:be an equivalence relation on the set
2957:Discrete mathematics and graph theory
1637:for universal algebra. Formally, let
7:
2939:. World Scientific. pp. 14–17.
2904:Keith Kearnes; Emil W. Kiss (2013).
558:{\displaystyle f_{i}^{\mathcal {A}}}
211:{\displaystyle (a_{i},\;b_{i})\in E}
46:. Quotient algebras are also called
467:Quotient algebras and homomorphisms
2486:
2483:
2480:
2334:
2331:
2328:
2307:
2304:
2301:
2280:
2277:
2274:
2194:
2191:
2188:
2123:, with respect to a fixed algebra
2016:
2013:
2010:
1989:
1986:
1983:
1962:
1959:
1956:
1887:
1884:
1881:
1777:
1774:
1771:
1712:
1709:
1706:
1606:
1603:
1600:
1383:
1380:
1377:
424:{\displaystyle a_{i},\;b_{i}\in A}
14:
2778:This line of research led to the
1019:denotes the equivalence class of
2906:The Shape of Congruence Lattices
1353:{\displaystyle {\mathcal {A}}/E}
2936:Universal algebra and coalgebra
2869:Canadian Mathematical Bulletin
2553:{\displaystyle {\mathcal {A}}}
2515:
2500:
2490:
2476:
2462:{\displaystyle {\mathcal {A}}}
2348:
2338:
2324:
2321:
2311:
2294:
2284:
2253:{\displaystyle {\mathcal {A}}}
2208:
2198:
2140:{\displaystyle {\mathcal {A}}}
2030:
2020:
2006:
2003:
1993:
1976:
1966:
1931:{\displaystyle {\mathcal {A}}}
1901:
1891:
1825:{\displaystyle {\mathcal {A}}}
1748:{\displaystyle {\mathcal {B}}}
1657:
1572:{\displaystyle {\mathcal {A}}}
1544:{\displaystyle {\mathcal {A}}}
1516:{\displaystyle {\mathcal {A}}}
1467:
1456:
1447:
1441:
1433:
1415:
1398:
1321:{\displaystyle {\mathcal {A}}}
1289:{\displaystyle {\mathcal {A}}}
1251:
1236:
1205:
1188:
1156:{\displaystyle {\mathcal {A}}}
1122:
1107:
1086:
1077:
986:
979:
953:
949:
910:
890:
884:
875:
854:
836:
822:
819:
761:
745:
730:
686:{\displaystyle {\mathcal {A}}}
636:{\displaystyle {\mathcal {A}}}
612:{\displaystyle {\mathcal {A}}}
512:{\displaystyle {\mathcal {A}}}
365:
362:
317:
308:
263:
257:
199:
172:
130:{\displaystyle {\mathcal {A}}}
1:
2954:Purna Chandra Biswal (2005).
456:{\displaystyle 1\leq i\leq n}
243:{\displaystyle 1\leq i\leq n}
669:enumerates the functions in
1012:{\displaystyle _{E}\in A/E}
3020:
2849:10.7146/math.scand.a-10850
2797:Congruence lattice problem
693:and their arities) define
18:
2975:Clifford Bergman (2011).
1859:{\displaystyle A\times A}
1635:first isomorphism theorem
1631:homomorphic image theorem
471:Any equivalence relation
101:into a common framework.
2836:Mathematica Scandinavica
2830:Jonnson, Bjarni (1967).
2749:= (x / (y \ y))(y \ z))
2574:composition of relations
1629:, a result known as the
2652:. An algebra is called
479:partitions this set in
2882:10.4153/CMB-1969-016-6
2780:Pixley–Wille algorithm
2646:
2608:
2554:
2522:
2463:
2436:
2355:
2254:
2230:
2141:
2103:
2037:
1932:
1908:
1860:
1826:
1790:
1749:
1725:
1671:
1619:
1573:
1545:
1517:
1490:
1354:
1322:
1290:
1258:
1157:
1129:
1039:
1038:{\displaystyle x\in A}
1013:
966:
779:
687:
663:
662:{\displaystyle i\in I}
637:
613:
589:
559:
513:
457:
425:
378:
244:
212:
131:
2753:complemented lattices
2655:congruence permutable
2647:
2609:
2555:
2523:
2464:
2437:
2356:
2255:
2231:
2142:
2104:
2038:
1933:
1909:
1861:
1827:
1791:
1750:
1726:
1672:
1620:
1574:
1546:
1518:
1491:
1368:of the homomorphism,
1355:
1323:
1300:. There is a natural
1291:
1259:
1158:
1135:, given a congruence
1130:
1040:
1014:
967:
780:
688:
664:
638:
614:
590:
588:{\displaystyle n_{i}}
560:
514:
458:
426:
379:
245:
213:
155:substitution property
132:
99:representation theory
54:that is additionally
2802:Lattice of subgroups
2618:
2580:
2540:
2473:
2449:
2365:
2264:
2240:
2151:
2127:
2047:
1946:
1918:
1877:
1844:
1812:
1766:
1735:
1689:
1641:
1583:
1559:
1531:
1503:
1372:
1332:
1308:
1276:
1167:
1143:
1064:
1023:
976:
789:
697:
673:
647:
643:, and the subscript
623:
599:
572:
535:
499:
435:
388:
254:
222:
169:
157:with respect to) an
117:
52:equivalence relation
2576:as operation, i.e.
2572:(commute) with the
2568:If two congruences
1234:
1105:
909:
818:
726:
554:
481:equivalence classes
105:Compatible relation
64:equivalence classes
44:congruence relation
40:algebraic structure
38:the elements of an
2863:Day, Alan (1969).
2642:
2604:
2564:Maltsev conditions
2550:
2534:congruence lattice
2518:
2459:
2445:For every algebra
2432:
2351:
2250:
2226:
2137:
2099:
2033:
1928:
1904:
1856:
1822:
1808:For every algebra
1804:Congruence lattice
1786:
1745:
1721:
1667:
1615:
1569:
1541:
1513:
1486:
1350:
1318:
1286:
1254:
1208:
1153:
1125:
1089:
1053: modulo
1035:
1009:
962:
893:
792:
775:
700:
683:
659:
633:
609:
585:
555:
538:
509:
453:
421:
374:
240:
208:
153:with (or have the
127:
3004:Universal algebra
2986:978-1-4398-5129-6
2967:978-81-203-2721-4
2946:978-981-283-745-5
2915:978-0-8218-8323-5
1838:identity relation
1523:, a homomorphism
1499:Given an algebra
495:. For an algebra
34:is the result of
3011:
2990:
2971:
2950:
2920:
2919:
2901:
2895:
2894:
2884:
2860:
2854:
2853:
2851:
2827:
2821:
2818:
2757:Heyting algebras
2750:
2739:
2726:
2687:
2651:
2649:
2648:
2643:
2613:
2611:
2610:
2605:
2559:
2557:
2556:
2551:
2549:
2548:
2527:
2525:
2524:
2519:
2499:
2498:
2489:
2468:
2466:
2465:
2460:
2458:
2457:
2441:
2439:
2438:
2433:
2431:
2430:
2429:
2419:
2418:
2406:
2405:
2390:
2389:
2377:
2376:
2360:
2358:
2357:
2352:
2347:
2346:
2337:
2320:
2319:
2310:
2293:
2292:
2283:
2259:
2257:
2256:
2251:
2249:
2248:
2235:
2233:
2232:
2227:
2207:
2206:
2197:
2171:
2170:
2169:
2146:
2144:
2143:
2138:
2136:
2135:
2108:
2106:
2105:
2100:
2098:
2097:
2085:
2084:
2072:
2071:
2059:
2058:
2042:
2040:
2039:
2034:
2029:
2028:
2019:
2002:
2001:
1992:
1975:
1974:
1965:
1937:
1935:
1934:
1929:
1927:
1926:
1913:
1911:
1910:
1905:
1900:
1899:
1890:
1865:
1863:
1862:
1857:
1831:
1829:
1828:
1823:
1821:
1820:
1795:
1793:
1792:
1787:
1781:
1780:
1754:
1752:
1751:
1746:
1744:
1743:
1730:
1728:
1727:
1722:
1716:
1715:
1703:
1698:
1697:
1676:
1674:
1673:
1668:
1666:
1665:
1656:
1655:
1624:
1622:
1621:
1616:
1610:
1609:
1597:
1592:
1591:
1578:
1576:
1575:
1570:
1568:
1567:
1550:
1548:
1547:
1542:
1540:
1539:
1522:
1520:
1519:
1514:
1512:
1511:
1495:
1493:
1492:
1487:
1485:
1484:
1466:
1436:
1430:
1429:
1414:
1387:
1386:
1359:
1357:
1356:
1351:
1346:
1341:
1340:
1327:
1325:
1324:
1319:
1317:
1316:
1295:
1293:
1292:
1287:
1285:
1284:
1266:quotient algebra
1263:
1261:
1260:
1255:
1250:
1249:
1233:
1229:
1224:
1223:
1216:
1198:
1181:
1176:
1175:
1162:
1160:
1159:
1154:
1152:
1151:
1134:
1132:
1131:
1126:
1121:
1120:
1104:
1103:
1097:
1073:
1072:
1044:
1042:
1041:
1036:
1018:
1016:
1015:
1010:
1005:
994:
993:
971:
969:
968:
963:
961:
960:
948:
947:
946:
945:
922:
921:
908:
907:
901:
883:
882:
873:
872:
871:
870:
844:
843:
834:
833:
817:
813:
808:
807:
800:
784:
782:
781:
776:
771:
760:
759:
758:
757:
740:
725:
721:
716:
715:
708:
692:
690:
689:
684:
682:
681:
668:
666:
665:
660:
642:
640:
639:
634:
632:
631:
618:
616:
615:
610:
608:
607:
594:
592:
591:
586:
584:
583:
564:
562:
561:
556:
553:
552:
546:
518:
516:
515:
510:
508:
507:
462:
460:
459:
454:
430:
428:
427:
422:
414:
413:
400:
399:
383:
381:
380:
375:
361:
360:
342:
341:
329:
328:
307:
306:
288:
287:
275:
274:
249:
247:
246:
241:
217:
215:
214:
209:
198:
197:
184:
183:
136:
134:
133:
128:
126:
125:
95:quotient modules
32:quotient algebra
3019:
3018:
3014:
3013:
3012:
3010:
3009:
3008:
2994:
2993:
2987:
2974:
2968:
2953:
2947:
2932:
2929:
2924:
2923:
2916:
2903:
2902:
2898:
2862:
2861:
2857:
2829:
2828:
2824:
2819:
2815:
2810:
2788:
2761:modular lattice
2745:
2728:
2689:
2670:
2667:Anatoly Maltsev
2616:
2615:
2578:
2577:
2566:
2538:
2537:
2471:
2470:
2447:
2446:
2420:
2410:
2397:
2381:
2368:
2363:
2362:
2262:
2261:
2238:
2237:
2160:
2149:
2148:
2125:
2124:
2118:binary relation
2089:
2076:
2063:
2050:
2045:
2044:
1944:
1943:
1916:
1915:
1875:
1874:
1842:
1841:
1810:
1809:
1806:
1764:
1763:
1733:
1732:
1687:
1686:
1639:
1638:
1581:
1580:
1557:
1556:
1529:
1528:
1501:
1500:
1476:
1459:
1421:
1407:
1370:
1369:
1330:
1329:
1306:
1305:
1274:
1273:
1235:
1165:
1164:
1141:
1140:
1106:
1062:
1061:
1060:For an algebra
1021:
1020:
985:
974:
973:
952:
937:
932:
913:
874:
862:
857:
835:
825:
787:
786:
749:
744:
695:
694:
671:
670:
645:
644:
621:
620:
597:
596:
575:
570:
569:
533:
532:
497:
496:
469:
433:
432:
405:
391:
386:
385:
352:
333:
320:
298:
279:
266:
252:
251:
220:
219:
189:
175:
167:
166:
161:-ary operation
145:. The relation
115:
114:
107:
87:quotient spaces
79:quotient groups
48:factor algebras
24:
17:
12:
11:
5:
3017:
3015:
3007:
3006:
2996:
2995:
2992:
2991:
2985:
2972:
2966:
2951:
2945:
2928:
2925:
2922:
2921:
2914:
2896:
2875:(2): 167–173.
2855:
2822:
2812:
2811:
2809:
2806:
2805:
2804:
2799:
2794:
2787:
2784:
2768:Bjarni Jónsson
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2565:
2562:
2547:
2517:
2514:
2511:
2508:
2505:
2502:
2497:
2492:
2488:
2485:
2482:
2478:
2456:
2428:
2423:
2417:
2413:
2409:
2404:
2400:
2396:
2393:
2388:
2384:
2380:
2375:
2371:
2350:
2345:
2340:
2336:
2333:
2330:
2326:
2323:
2318:
2313:
2309:
2306:
2303:
2299:
2296:
2291:
2286:
2282:
2279:
2276:
2272:
2269:
2247:
2225:
2222:
2219:
2216:
2213:
2210:
2205:
2200:
2196:
2193:
2190:
2186:
2183:
2180:
2177:
2174:
2168:
2163:
2159:
2156:
2134:
2096:
2092:
2088:
2083:
2079:
2075:
2070:
2066:
2062:
2057:
2053:
2032:
2027:
2022:
2018:
2015:
2012:
2008:
2005:
2000:
1995:
1991:
1988:
1985:
1981:
1978:
1973:
1968:
1964:
1961:
1958:
1954:
1951:
1940:meet operation
1925:
1903:
1898:
1893:
1889:
1886:
1883:
1855:
1852:
1849:
1819:
1805:
1802:
1785:
1779:
1776:
1773:
1742:
1720:
1714:
1711:
1708:
1702:
1696:
1664:
1659:
1654:
1649:
1646:
1614:
1608:
1605:
1602:
1596:
1590:
1566:
1538:
1510:
1483:
1479:
1475:
1472:
1469:
1465:
1462:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1428:
1424:
1420:
1417:
1413:
1410:
1406:
1403:
1400:
1397:
1394:
1391:
1385:
1382:
1379:
1349:
1345:
1339:
1315:
1283:
1270:factor algebra
1264:is called the
1253:
1248:
1245:
1242:
1238:
1232:
1228:
1222:
1215:
1211:
1207:
1204:
1201:
1197:
1193:
1190:
1187:
1184:
1180:
1174:
1163:, the algebra
1150:
1124:
1119:
1116:
1113:
1109:
1102:
1096:
1092:
1088:
1085:
1082:
1079:
1076:
1071:
1034:
1031:
1028:
1008:
1004:
1000:
997:
992:
988:
984:
981:
959:
955:
951:
944:
940:
935:
931:
928:
925:
920:
916:
912:
906:
900:
896:
892:
889:
886:
881:
877:
869:
865:
860:
856:
853:
850:
847:
842:
838:
832:
828:
824:
821:
816:
812:
806:
799:
795:
774:
770:
766:
763:
756:
752:
747:
743:
739:
735:
732:
729:
724:
720:
714:
707:
703:
680:
658:
655:
652:
630:
606:
582:
578:
551:
545:
541:
506:
487:, and denoted
468:
465:
452:
449:
446:
443:
440:
420:
417:
412:
408:
403:
398:
394:
373:
370:
367:
364:
359:
355:
351:
348:
345:
340:
336:
332:
327:
323:
319:
316:
313:
310:
305:
301:
297:
294:
291:
286:
282:
278:
273:
269:
265:
262:
259:
239:
236:
233:
230:
227:
207:
204:
201:
196:
192:
187:
182:
178:
174:
149:is said to be
124:
106:
103:
91:linear algebra
71:quotient rings
15:
13:
10:
9:
6:
4:
3:
2:
3016:
3005:
3002:
3001:
2999:
2988:
2982:
2978:
2973:
2969:
2963:
2959:
2958:
2952:
2948:
2942:
2938:
2937:
2931:
2930:
2926:
2917:
2911:
2907:
2900:
2897:
2892:
2888:
2883:
2878:
2874:
2870:
2866:
2859:
2856:
2850:
2845:
2841:
2837:
2833:
2826:
2823:
2817:
2814:
2807:
2803:
2800:
2798:
2795:
2793:
2792:Quotient ring
2790:
2789:
2785:
2783:
2781:
2776:
2773:
2769:
2764:
2762:
2758:
2754:
2748:
2743:
2738:
2735:
2731:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2685:
2681:
2677:
2673:
2668:
2663:
2661:
2657:
2656:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2575:
2571:
2563:
2561:
2535:
2532:, called the
2531:
2512:
2509:
2506:
2503:
2443:
2415:
2411:
2407:
2402:
2398:
2391:
2386:
2382:
2378:
2373:
2369:
2297:
2270:
2267:
2220:
2217:
2214:
2211:
2184:
2181:
2175:
2172:
2157:
2122:
2119:
2115:
2110:
2094:
2090:
2086:
2081:
2077:
2073:
2068:
2064:
2060:
2055:
2051:
1979:
1952:
1949:
1941:
1871:
1869:
1853:
1850:
1847:
1839:
1835:
1803:
1801:
1799:
1783:
1761:
1758:
1718:
1700:
1684:
1680:
1647:
1644:
1636:
1632:
1628:
1612:
1594:
1554:
1526:
1497:
1481:
1477:
1473:
1463:
1460:
1453:
1450:
1444:
1438:
1426:
1422:
1418:
1411:
1408:
1404:
1401:
1392:
1389:
1367:
1363:
1347:
1343:
1303:
1299:
1271:
1267:
1246:
1243:
1240:
1230:
1226:
1213:
1209:
1202:
1199:
1195:
1191:
1185:
1182:
1178:
1138:
1117:
1114:
1111:
1094:
1090:
1083:
1080:
1074:
1058:
1056:
1052:
1048:
1045:generated by
1032:
1029:
1026:
1006:
1002:
998:
995:
990:
982:
957:
942:
938:
933:
929:
926:
923:
918:
914:
898:
894:
887:
879:
867:
863:
858:
851:
848:
845:
840:
830:
826:
814:
810:
797:
793:
772:
768:
764:
754:
750:
741:
737:
733:
727:
722:
718:
705:
701:
656:
653:
650:
580:
576:
568:
543:
539:
530:
526:
522:
494:
490:
486:
482:
478:
474:
466:
464:
450:
447:
444:
441:
438:
418:
415:
410:
406:
401:
396:
392:
371:
368:
357:
353:
349:
346:
343:
338:
334:
330:
325:
321:
314:
311:
303:
299:
295:
292:
289:
284:
280:
276:
271:
267:
260:
237:
234:
231:
228:
225:
205:
202:
194:
190:
185:
180:
176:
164:
160:
156:
152:
148:
144:
140:
112:
104:
102:
100:
96:
92:
88:
84:
80:
76:
72:
67:
65:
61:
58:with all the
57:
53:
49:
45:
41:
37:
33:
29:
22:
21:quotient ring
2976:
2956:
2935:
2905:
2899:
2872:
2868:
2858:
2839:
2835:
2825:
2816:
2777:
2765:
2746:
2736:
2733:
2729:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2683:
2679:
2675:
2671:
2664:
2653:
2569:
2567:
2533:
2444:
2120:
2111:
1872:
1867:
1833:
1807:
1797:
1756:
1682:
1630:
1625:The two are
1524:
1498:
1361:
1302:homomorphism
1297:
1269:
1265:
1136:
1059:
1054:
1050:
1046:
528:
524:
520:
492:
488:
485:quotient set
476:
472:
470:
162:
158:
154:
150:
146:
142:
138:
110:
108:
83:group theory
68:
55:
47:
36:partitioning
31:
25:
2742:quasigroups
1832:on the set
75:ring theory
28:mathematics
2927:References
2772:conditions
2770:found the
2740:), rings,
2688:such that
1840:on A, and
1755:such that
1679:surjective
1633:or as the
1627:isomorphic
151:compatible
137:, and let
60:operations
56:compatible
2891:120602601
2665:In 1954,
2640:β
2637:∨
2634:α
2628:β
2625:∘
2622:α
2602:α
2599:∘
2596:β
2590:β
2587:∘
2584:α
2513:∨
2507:∧
2422:⟩
2408:∪
2395:⟨
2379:∨
2325:→
2298:×
2268:∨
2218:⊆
2212:∣
2185:∈
2176:⋂
2162:⟩
2155:⟨
2087:∩
2061:∧
2007:→
1980:×
1950:∧
1851:×
1658:→
1474:⊆
1419:∈
1244:∈
1115:∈
1030:∈
996:∈
927:…
849:…
762:→
654:∈
475:in a set
448:≤
442:≤
416:∈
369:∈
347:…
293:…
235:≤
229:≤
203:∈
2998:Category
2786:See also
1760:composed
1464:′
1412:′
972:, where
384:for any
250:implies
93:and the
42:using a
2842:: 110.
2660:variety
2570:permute
2530:lattice
2116:of any
2114:closure
1796:equals
1296:modulo
2983:
2964:
2943:
2912:
2889:
2744:(take
1868:simple
1836:, the
1579:) and
1551:, the
1366:kernel
85:, the
2887:S2CID
2808:Notes
1731:onto
1685:from
1677:be a
1553:image
1304:from
1272:) of
567:arity
431:with
165:, if
2981:ISBN
2962:ISBN
2941:ISBN
2910:ISBN
2705:) ≈
1873:Let
1268:(or
1057:").
218:for
109:Let
30:, a
2877:doi
2844:doi
2536:of
2361:as
1328:to
1139:on
785:as
595:in
565:of
527:if
97:of
89:of
81:of
73:of
26:In
3000::
2885:.
2873:12
2871:.
2867:.
2840:21
2838:.
2834:.
2755:,
2751:,
2734:xy
2732:=
2721:,
2717:,
2709:≈
2701:,
2697:,
2682:,
2678:,
2560:.
2469:,
2442:.
2109:.
1942::
1870:.
1800:.
1555:h(
1496:.
1049:("
77:,
2989:.
2970:.
2949:.
2918:.
2893:.
2879::
2852:.
2846::
2747:q
2737:z
2730:q
2725:)
2723:x
2719:y
2715:y
2713:(
2711:q
2707:x
2703:y
2699:y
2695:x
2693:(
2691:q
2686:)
2684:z
2680:y
2676:x
2674:(
2672:q
2631:=
2593:=
2546:A
2516:)
2510:,
2504:,
2501:)
2496:A
2491:(
2487:n
2484:o
2481:C
2477:(
2455:A
2427:A
2416:2
2412:E
2403:1
2399:E
2392:=
2387:2
2383:E
2374:1
2370:E
2349:)
2344:A
2339:(
2335:n
2332:o
2329:C
2322:)
2317:A
2312:(
2308:n
2305:o
2302:C
2295:)
2290:A
2285:(
2281:n
2278:o
2275:C
2271::
2246:A
2224:}
2221:F
2215:E
2209:)
2204:A
2199:(
2195:n
2192:o
2189:C
2182:F
2179:{
2173:=
2167:A
2158:E
2133:A
2121:E
2095:2
2091:E
2082:1
2078:E
2074:=
2069:2
2065:E
2056:1
2052:E
2031:)
2026:A
2021:(
2017:n
2014:o
2011:C
2004:)
1999:A
1994:(
1990:n
1987:o
1984:C
1977:)
1972:A
1967:(
1963:n
1960:o
1957:C
1953::
1924:A
1902:)
1897:A
1892:(
1888:n
1885:o
1882:C
1854:A
1848:A
1834:A
1818:A
1798:h
1784:h
1778:r
1775:e
1772:k
1757:g
1741:B
1719:h
1713:r
1710:e
1707:k
1701:/
1695:A
1683:g
1663:B
1653:A
1648::
1645:h
1613:h
1607:r
1604:e
1601:k
1595:/
1589:A
1565:A
1537:A
1525:h
1509:A
1482:2
1478:A
1471:}
1468:)
1461:a
1457:(
1454:h
1451:=
1448:)
1445:a
1442:(
1439:h
1434:|
1427:2
1423:A
1416:)
1409:a
1405:,
1402:a
1399:(
1396:{
1393:=
1390:h
1384:r
1381:e
1378:k
1362:h
1348:E
1344:/
1338:A
1314:A
1298:E
1282:A
1252:)
1247:I
1241:i
1237:)
1231:E
1227:/
1221:A
1214:i
1210:f
1206:(
1203:,
1200:E
1196:/
1192:A
1189:(
1186:=
1183:E
1179:/
1173:A
1149:A
1137:E
1123:)
1118:I
1112:i
1108:)
1101:A
1095:i
1091:f
1087:(
1084:,
1081:A
1078:(
1075:=
1070:A
1055:E
1051:x
1047:E
1033:A
1027:x
1007:E
1003:/
999:A
991:E
987:]
983:x
980:[
958:E
954:]
950:)
943:i
939:n
934:a
930:,
924:,
919:1
915:a
911:(
905:A
899:i
895:f
891:[
888:=
885:)
880:E
876:]
868:i
864:n
859:a
855:[
852:,
846:,
841:E
837:]
831:1
827:a
823:[
820:(
815:E
811:/
805:A
798:i
794:f
773:E
769:/
765:A
755:i
751:n
746:)
742:E
738:/
734:A
731:(
728::
723:E
719:/
713:A
706:i
702:f
679:A
657:I
651:i
629:A
605:A
581:i
577:n
550:A
544:i
540:f
529:E
525:E
523:/
521:A
505:A
493:E
491:/
489:A
477:A
473:E
451:n
445:i
439:1
419:A
411:i
407:b
402:,
397:i
393:a
372:E
366:)
363:)
358:n
354:b
350:,
344:,
339:2
335:b
331:,
326:1
322:b
318:(
315:f
312:,
309:)
304:n
300:a
296:,
290:,
285:2
281:a
277:,
272:1
268:a
264:(
261:f
258:(
238:n
232:i
226:1
206:E
200:)
195:i
191:b
186:,
181:i
177:a
173:(
163:f
159:n
147:E
143:A
139:E
123:A
111:A
23:.
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