1383:
1111:
1099:
500:
1165:
24:
1141:. The example shows that this inequality can be strict in general. In a modular lattice, however, equality holds. Since the dual of a modular lattice is again modular, φψ is also the identity on , and therefore the two maps φ and ψ are isomorphisms between these two intervals. This result is sometimes called the
1413:
In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility. In a digression he introduced and studied lattices formally in a general context. He observed that the lattice of submodules of a module satisfies the
868:
1277:. Thus, in an M-symmetric lattice, every right modular element is also left modular, and vice-versa. Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. In the lattice
1791:
1335:
if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M-symmetric. The same equivalence holds for infinite lattices which satisfy the
745:
1472:
A paper published by
Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture).
1098:
1354:) is dually modular. Cross-symmetry implies M-symmetry but not M-symmetry. Therefore, cross-symmetry is not equivalent to dual cross-symmetry. A lattice with a least element 0 is
1835:
2208:
699:
proved that, in every finite modular lattice, the number of join-irreducible elements equals the number of meet-irreducible elements. More generally, for every
1686:
1410:" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual.
1110:
2114:
131:
respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the
2165:
2015:
2023:
Festschrift der
Herzogl. Technischen Hochschule Carolo-Wilhelmina bei Gelegenheit der 69. Versammlung Deutscher Naturforscher und Ärzte in Braunschweig
1837:, is true for any lattice. Substituting this for the second conjunct of the right-hand side of the former equation yields the Modular Identity.
2302:
2276:
2086:
1904:
1668:
1631:
2255:
1394:, who published most of the relevant papers after his retirement. In a paper published in 1894 he studied lattices, which he called
1427:
In the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual:
1145:
for modular lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair of elements.
875:
Sketch of proof: Let G be modular, and let the premise of the implication hold. Then using absorption and modular identity:
863:{\displaystyle {\Big (}(c\leq a){\text{ and }}(a\wedge b=c\wedge b){\text{ and }}(a\vee b=c\vee b){\Big )}\Rightarrow (a=c)}
2196:
2106:
1887:
Bogart, Kenneth P.; Freese, Ralph; Kung, Joseph P. S., eds. (1990), "Proof of a
Conjecture on Finite Modular Lattices",
1487:
1382:
2191:
2101:
582:
of a group is not modular. For an example, the lattice of subgroups of the dihedral group of order 8 is not modular.
484:
136:
135:. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a
2294:
1121:
The composition ψφ is an order-preserving map from the interval to itself which also satisfies the inequality ψ(φ(
289:. In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.
151:
1336:
2359:
1988:
and the converse holds for lattices of finite length, this can only lead to confusion for infinite lattices.
2096:
1328:
59:
1796:
2346:
An open-source browser-based web application that can generate and visualize some free modular lattices.
1852:
1498:
1469:. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.
1053:
of a modular lattice, one can consider the intervals and . They are connected by order-preserving maps
2186:
1493:
1466:
1407:
690:
579:
2271:, IAS/Park City Mathematics Series, vol. 13, American Mathematical Society, pp. 389–496,
1985:
1149:
575:
212:
2264:
2217:
2057:
1869:
1847:
1104:
In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.
560:
378:
The modular law can be expressed as an equation that is required to hold unconditionally. Since
159:
150:
and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the
17:
1971:, Definition 4.12)). These notions are equivalent in a semimodular lattice, but not in general.
1320:, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.
1037:
must hold. The rest of the proof is routine manipulation with infima, suprema and inequalities.
499:
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2272:
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2174:
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1900:
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that connects the two lattice operations similarly to the way in which the associative law λ(μ
140:
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2011:
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28:
1786:{\displaystyle (a\wedge b)\vee (x\wedge b)=((a\wedge b)\vee x)\wedge ((a\wedge b)\vee b)}
2287:
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2134:
2061:
1951:
Modular element has been varying defined by different authors to mean right modular (
1503:
1481:
1230:, i.e. if one half of the diamond isomorphism theorem holds for the pair. An element
564:
239:
128:
124:
32:
2242:, de Gruyter Expositions in Mathematics, vol. 14, Walter de Gruyter & Co.,
211:, and there are various generalizations of modularity related to this notion and to
2157:
1648:
1611:
674:
holds, contradicting the modular law. Every non-modular lattice contains a copy of
155:
47:
2231:
250:
for vector spaces connects multiplication in the field and scalar multiplication.
23:
2141:, Grundlehren der mathematischen Wissenschaften, vol. 300, Springer-Verlag,
1896:
2203:
2130:
1148:
The diamond isomorphism theorem for modular lattices is analogous to the second
2323:
2146:
488:
2178:
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1623:
937:
For the other direction, let the implication of the theorem hold in G. Let
2247:
1959:, Definition 2.25)), both left and right modular (or dual right modular) (
1522:
1424:). He also proved that the modular identity and its dual are equivalent.
2318:
1484:, a class of graphs that includes the Hasse diagrams of modular lattices
2053:
1873:
714:
A useful property to show that a lattice is not modular is as follows:
147:
2343:
2335:
sequence A006981 (Number of unlabeled modular lattices with
1490:, an infinite modular lattice defined on strings of the digits 1 and 2
1342:
Several less important notions are also closely related. A lattice is
169:
for which the modular law holds in connection with arbitrary elements
2222:
2016:"Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler"
1891:, Contemporary Mathematicians, Boston: Birkhäuser, pp. 219–224,
1865:
1116:
Failure of the diamond isomorphism theorem in a non-modular lattice.
328:
in every lattice. Therefore, the modular law can also be stated as
1381:
1163:
498:
165:
In a not necessarily modular lattice, there may still be elements
22:
2331:
1465:). In modern literature, they are more commonly referred to as
563:
is modular. As a special case, the lattice of subgroups of an
1980:
Some authors, e.g. Fofanova (2001), refer to such lattices as
707:
other elements equals the number that are covered by exactly
2115:"On the symmetry of the modular relation in atomic lattices"
1889:
The
Dilworth Theorems: Selected Papers of Robert P. Dilworth
1850:(1954), "Proof of a conjecture on finite modular lattices",
123:
are arbitrary elements in the lattice, ≤ is the
2334:
703:, the number of elements of the lattice that cover exactly
585:
The smallest non-modular lattice is the "pentagon" lattice
491:
and direct products of modular lattices are again modular.
1967:, p. 43)), or satisfying a modular rank condition (
1386:
Free modular lattice generated by three elements {x,y,z}
989:). From the modular inequality immediately follows that
191:. Even more generally, the modular law may hold for any
1331:
lattice, and a lattice is called dually M-symmetric or
432:
in the defining equation of the modular law to obtain:
2267:(2007), "An Introduction to Hyperplane Arrangements",
2079:
Modern algebra and the rise of mathematical structures
2206:(1999), "Why the characteristic polynomial factors",
1799:
1689:
748:
2286:
1829:
1785:
862:
31:2. As with all finite 2-dimensional lattices, its
837:
751:
2006:
2004:
2002:
2000:
1998:
1996:
1994:
1683:In a distributive lattice, the following holds:
1299:is not M-symmetric. The centred hexagon lattice
487:of lattices. Therefore, all homomorphic images,
483:, the modular lattices form a subvariety of the
1618:. Universitext. London: Springer. Theorem 4.4.
2119:Journal of Science of the Hiroshima University
2038:"Über die von drei Moduln erzeugte Dualgruppe"
1655:. Universitext. London: Springer. p. 65.
1455:He called lattices that satisfy this identity
1152:in algebra, and it is a generalization of the
2209:Bulletin of the American Mathematical Society
127:, and ∨ and ∧ (called
8:
2166:Notices of the American Mathematical Society
1460:
1419:
263:is clearly necessary, since it follows from
238:The modular law can be seen as a restricted
1956:
1414:modular identity. He called such lattices
2221:
1798:
1688:
1653:Lattices and Ordered Algebraic Structures
1616:Lattices and Ordered Algebraic Structures
836:
835:
803:
771:
750:
749:
747:
226:, who discovered the modular identity in
1323:M-symmetry is not a self-dual notion. A
696:
479:This shows that, using terminology from
158:(and more generally the submodules of a
1968:
1964:
1545:The following is true for any lattice:
1514:
1390:The definition of modularity is due to
1094:
1306:is M-symmetric but not modular. Since
1273:) is also a modular pair is called an
218:Modular lattices are sometimes called
1984:. Since every M-symmetric lattice is
1960:
1952:
1523:"Why are modular lattices important?"
1339:(or the descending chain condition).
1265:A lattice with the property that if (
1258:) is a modular pair for all elements
1242:) is a modular pair for all elements
7:
509:, the smallest non-modular lattice:
146:Modular lattices arise naturally in
46:In the branch of mathematics called
1830:{\displaystyle (a\wedge b)\vee b=b}
722:is modular if and only if, for any
154:. For example, the subspaces of a
2158:"The many lives of lattice theory"
2081:(2nd ed.), pp. 121–129,
1327:is a pair which is modular in the
592:consisting of five elements 0, 1,
58:that satisfies the following self-
14:
1182:, is M-symmetric but not modular.
1160:Modular pairs and related notions
2185:Skornyakov, L. A. (2001) ,
1793:. Moreover, the absorption law,
1194:) of elements such that for all
1109:
1097:
949:be any elements in G, such that
578:is modular. But in general the
559:The lattice of submodules of a
187:). Such an element is called a
2344:Free Modular Lattice Generator
2095:Fofanova, T. S. (2001) ,
1955:, p. 74)), left modular (
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1406:) as part of his "algebra of
1897:10.1007/978-1-4899-3558-8_21
1288:) is modular, but the pair (
1168:The centred hexagon lattice
1029:, then using the assumption
227:
2240:Subgroup lattices of groups
2192:Encyclopedia of Mathematics
2139:Arrangements of Hyperplanes
2102:Encyclopedia of Mathematics
2025:, Friedrich Vieweg und Sohn
1358:if for every modular pair (
1346:if for every modular pair (
1284:described above, the pair (
1269:) is a modular pair, then (
1143:diamond isomorphism theorem
1041:Diamond isomorphism theorem
228:several motivating examples
133:diamond isomorphism theorem
2376:
2295:Cambridge University Press
2036:Dedekind, Richard (1900),
1527:Mathematics Stack Exchange
1462:Dualgruppen vom Idealtypus
1421:Dualgruppen vom Modultypus
1416:dual groups of module type
207:. Such a pair is called a
162:) form a modular lattice.
15:
2147:10.1007/978-3-662-02772-1
2113:Maeda, Shûichirô (1965),
2077:Corry, Leo (2003-11-27),
1922:term for modular pair is
1457:dual groups of ideal type
1370: = 0 the pair (
1337:ascending chain condition
1234:of a lattice is called a
2238:Schmidt, Roland (1994),
580:lattice of all subgroups
16:Not to be confused with
2285:Stern, Manfred (1999),
2269:Geometric combinatorics
1957:Orlik & Terao (1992
1661:10.1007/1-84628-127-X_4
1624:10.1007/1-84628-127-X_4
1488:Young–Fibonacci lattice
292:It is easy to see that
152:2nd Isomorphism Theorem
2097:"Semi-modular lattice"
1831:
1787:
1461:
1420:
1403:
1387:
1183:
1068:that are defined by φ(
864:
556:
43:
2248:10.1515/9783110868647
2042:Mathematische Annalen
1853:Annals of Mathematics
1832:
1788:
1647:Blyth, T. S. (2005).
1610:Blyth, T. S. (2005).
1499:Supersolvable lattice
1467:distributive lattices
1385:
1292:) is not. Therefore,
1236:right modular element
1167:
1045:For any two elements
865:
620:is not comparable to
502:
333:Modular law (variant)
189:right modular element
27:A modular lattice of
26:
2289:Semimodular lattices
1982:semimodular lattices
1942:) are modular pairs.
1797:
1687:
1494:Orthomodular lattice
1252:left modular element
746:
691:distributive lattice
628:. For this lattice,
2265:Stanley, Richard P.
1934:in French if both (
1374:) is also modular.
1313:is a sublattice of
1275:M-symmetric lattice
1150:isomorphism theorem
2054:10.1007/BF01448979
1827:
1783:
1649:"Modular lattices"
1612:"Modular lattices"
1388:
1186:In any lattice, a
1184:
997:. If we show that
860:
561:module over a ring
557:
160:module over a ring
44:
18:unimodular lattice
2319:"Modular lattice"
2304:978-0-521-46105-4
2278:978-0-8218-3736-8
2187:"Modular lattice"
2173:(11): 1440–1445,
2088:978-3-7643-7002-2
2012:Dedekind, Richard
1906:978-1-4899-3560-1
1856:, Second Series,
1670:978-1-85233-905-0
1633:978-1-85233-905-0
1576:. Also, whenever
1325:dual modular pair
1246:, and an element
806:
774:
681:as a sublattice.
604:such that 0 <
481:universal algebra
220:Dedekind lattices
195:and a fixed pair
141:universal algebra
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1885:. Reprinted in
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1848:Dilworth, R. P.
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1524:
1518:
1515:
1509:
1505:
1504:Iwasawa group
1502:
1500:
1497:
1495:
1492:
1489:
1486:
1483:
1482:Modular graph
1480:
1479:
1475:
1473:
1470:
1468:
1463:
1458:
1450:
1446:
1442:
1438:
1434:
1430:
1429:
1428:
1425:
1422:
1417:
1411:
1409:
1405:
1401:
1397:
1393:
1384:
1377:
1375:
1373:
1369:
1366: ∧
1365:
1362:) satisfying
1361:
1357:
1353:
1349:
1345:
1340:
1338:
1334:
1330:
1326:
1321:
1316:
1309:
1302:
1295:
1291:
1287:
1280:
1276:
1272:
1268:
1263:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1229:
1226: =
1225:
1221:
1218: ∨
1217:
1213:
1210: ≤
1209:
1205:
1202: ∧
1201:
1197:
1193:
1189:
1178:
1171:
1166:
1159:
1157:
1155:
1151:
1146:
1144:
1140:
1136:
1132:
1128:
1124:
1112:
1107:
1100:
1095:
1093:
1091:
1087:
1083:
1079:
1075:
1071:
1061:
1058:
1057:
1056:
1055:
1054:
1052:
1048:
1040:
1038:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
933:
929:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
881:
878:
877:
876:
854:
851:
848:
829:
826:
823:
820:
817:
814:
811:
797:
794:
791:
788:
785:
782:
779:
765:
762:
759:
742:
741:
738:
734:
730:
726:
717:
716:
715:
712:
698:
694:
692:
684:
682:
677:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
631:
630:
629:
627:
623:
619:
615:
611:
607:
603:
599:
595:
588:
583:
581:
577:
573:
568:
566:
565:abelian group
562:
553:
549:
545:
539:
535:
531:
527:
521:
517:
513:
505:
501:
494:
492:
490:
486:
482:
473:
469:
465:
461:
457:
453:
449:
445:
440:
438:
435:
434:
433:
430:
426:
420:
414:
410:
406:
400:
396:
392:
386:
382:
371:
367:
363:
359:
355:
351:
344:
340:
336:
334:
331:
330:
329:
326:
322:
318:
314:
310:
306:
300:
296:
290:
287:
283:
279:
275:
271:
267:
261:
257:
251:
249:
245:
241:
233:
231:
229:
225:
221:
216:
214:
210:
204:
200:
190:
185:
181:
163:
161:
157:
153:
149:
144:
142:
138:
134:
130:
129:join and meet
126:
125:partial order
121:
117:
113:
104:
100:
96:
92:
88:
84:
78:
74:
70:
68:
65:
64:
63:
61:
57:
53:
49:
41:
40:-planar graph
39:
34:
33:Hasse diagram
30:
25:
19:
2336:
2322:
2288:
2268:
2239:
2223:math/9812136
2213:
2207:
2204:Sagan, Bruce
2190:
2170:
2164:
2138:
2131:Orlik, Peter
2122:
2118:
2100:
2078:
2045:
2041:
2031:
2022:
1981:
1976:
1961:Sagan (1999)
1947:
1939:
1935:
1931:
1927:
1923:
1914:
1888:
1857:
1851:
1842:
1679:
1652:
1642:
1615:
1605:
1596:
1592:
1588:
1582:
1578:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1541:
1530:. Retrieved
1526:
1517:
1471:
1456:
1454:
1448:
1444:
1440:
1436:
1432:
1426:
1415:
1412:
1395:
1389:
1371:
1367:
1363:
1359:
1355:
1351:
1350:) the pair (
1347:
1343:
1341:
1332:
1324:
1322:
1314:
1307:
1300:
1293:
1289:
1285:
1278:
1274:
1270:
1266:
1264:
1259:
1255:
1251:
1250:is called a
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1188:modular pair
1187:
1185:
1176:
1169:
1147:
1142:
1138:
1134:
1130:
1126:
1122:
1120:
1089:
1085:
1081:
1077:
1073:
1069:
1067:
1050:
1046:
1044:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
958:
954:
950:
946:
942:
938:
936:
931:
927:
923:
919:
915:
911:
907:
903:
899:
895:
891:
887:
883:
879:
874:
736:
732:
728:
724:
713:
695:
693:is modular.
688:
675:
673:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
625:
621:
617:
616:< 1, and
613:
609:
605:
601:
597:
593:
586:
584:
569:
567:is modular.
558:
551:
547:
543:
537:
533:
529:
525:
519:
515:
511:
503:
478:
471:
467:
463:
459:
455:
451:
447:
443:
436:
428:
424:
418:
412:
408:
404:
398:
394:
390:
384:
380:
377:
369:
365:
361:
357:
353:
349:
342:
338:
332:
324:
320:
316:
312:
308:
304:
298:
294:
291:
285:
281:
277:
273:
269:
265:
259:
255:
252:
247:
243:
237:
234:Introduction
219:
217:
209:modular pair
208:
202:
198:
188:
183:
179:
164:
156:vector space
145:
132:
119:
115:
111:
108:
102:
98:
94:
90:
86:
82:
76:
72:
66:
51:
48:order theory
45:
37:
1986:semimodular
1953:Stern (1999
1404:Dualgruppen
1396:dual groups
1356:⊥-symmetric
1333:M-symmetric
1214:, we have (
1198:satisfying
1190:is a pair (
489:sublattices
67:Modular law
62:condition,
2324:PlanetMath
2071:References
1926:. A pair (
1532:2018-09-17
1059:φ: → and
718:A lattice
685:Properties
416:, replace
402:and since
2339:elements)
2197:EMS Press
2179:0002-9920
2125:: 165–170
2107:EMS Press
2062:122529830
1940:b, a
1936:a, b
1928:a, b
1816:∨
1807:∧
1775:∨
1766:∧
1754:∧
1745:∨
1736:∧
1715:∧
1706:∨
1697:∧
1372:b, a
1360:a, b
1352:b, a
1348:a, b
1290:a, b
1286:b, a
1271:b, a
1267:a, b
1256:a, b
1240:a, b
1192:a, b
843:⇒
827:∨
815:∨
795:∧
783:∧
763:≤
2354:Category
2156:(1997),
2137:(1992),
2014:(1897),
1476:See also
735:∈
495:Examples
388:implies
346:implies
302:implies
246:) = (λμ)
80:implies
1938:) and (
1882:0063348
1874:1969639
1586:, then
1447:) = ∧
1408:modules
1378:History
485:variety
148:algebra
137:variety
56:lattice
2301:
2275:
2254:
2177:
2085:
2060:
1920:French
1903:
1880:
1872:
1667:
1630:
1400:German
1125:)) = (
1080:and ψ(
1062:ψ: →
957:. Let
689:Every
655:= 1 ∧
647:∨ 0 =
624:or to
458:) = ((
222:after
109:where
35:is an
2218:arXiv
2161:(PDF)
2058:S2CID
2019:(PDF)
1870:JSTOR
1566:) ∧ (
1558:) ≤ (
1510:Notes
1439:) ∨ (
651:<
608:<
576:group
574:of a
528:∨0 =
450:) ∨ (
422:with
315:) ≤ (
276:) = (
177:(for
93:) = (
54:is a
2332:OEIS
2299:ISBN
2273:ISBN
2252:ISBN
2175:ISSN
2083:ISBN
1918:The
1901:ISBN
1665:ISBN
1628:ISBN
1329:dual
1254:if (
1238:if (
1133:) ∧
1084:) =
1072:) =
969:) ∨
930:) =
918:) =
902:) ∨
890:) ∨
667:) ∧
643:) =
536:= 1∧
470:) ∧
466:) ∨
356:) ∧
323:) ∧
284:) ∧
173:and
101:) ∧
60:dual
50:, a
2244:doi
2228:doi
2143:doi
2050:doi
1893:doi
1862:doi
1657:doi
1620:doi
1550:∨ (
981:∧ (
961:= (
922:∧ (
910:∧ (
894:= (
882:= (
659:= (
635:∨ (
364:∨ (
307:∨ (
268:∨ (
85:∨ (
2356::
2321:.
2297:,
2293:,
2250:,
2226:,
2214:36
2212:,
2195:,
2189:,
2171:44
2169:,
2163:,
2133:;
2123:29
2121:,
2117:,
2105:,
2099:,
2056:,
2046:53
2044:,
2040:,
2021:,
1993:^
1963:,
1899:,
1878:MR
1876:,
1868:,
1858:60
1663:.
1651:.
1626:.
1614:.
1595:=
1591:∨
1581:≤
1570:∨
1562:∨
1554:∧
1525:.
1443:∧
1435:∧
1402::
1262:.
1206:≤
1156:.
1137:≥
1129:∨
1092:.
1088:∧
1076:∨
1033:=
1021:=
1013:,
1005:=
993:≤
977:=
973:,
953:≤
906:=
740:,
731:,
727:,
663:∨
639:∧
600:,
596:,
550:)∧
532:≠
524:=
514:∨(
462:∧
454:∧
446:∧
427:∧
411:≤
407:∧
397:∧
393:=
383:≤
368:∧
360:≤
352:∨
341:≤
319:∨
311:∧
297:≤
280:∨
272:∧
258:≤
230:.
215:.
201:,
182:≤
143:.
118:,
114:,
97:∨
89:∧
75:≤
38:st
2337:n
2327:.
2246::
2230::
2220::
2145::
2052::
1895::
1864::
1825:b
1822:=
1819:b
1813:)
1810:b
1804:a
1801:(
1781:)
1778:b
1772:)
1769:b
1763:a
1760:(
1757:(
1751:)
1748:x
1742:)
1739:b
1733:a
1730:(
1727:(
1724:=
1721:)
1718:b
1712:x
1709:(
1703:)
1700:b
1694:a
1691:(
1673:.
1659::
1636:.
1622::
1600:.
1597:b
1593:b
1589:a
1583:b
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1564:x
1560:a
1556:b
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1535:.
1459:(
1451:.
1449:b
1445:b
1441:a
1437:b
1433:x
1431:(
1418:(
1398:(
1368:b
1364:a
1318:7
1315:S
1311:5
1308:N
1304:7
1301:S
1297:5
1294:N
1282:5
1279:N
1260:b
1248:a
1244:a
1232:b
1228:x
1224:b
1220:a
1216:x
1212:b
1208:x
1204:b
1200:a
1196:x
1180:2
1177:D
1173:7
1170:S
1139:x
1135:b
1131:a
1127:x
1123:x
1090:b
1086:y
1082:y
1078:a
1074:x
1070:x
1051:b
1049:,
1047:a
1035:y
1031:x
1027:b
1025:∨
1023:y
1019:b
1017:∨
1015:x
1011:b
1009:∧
1007:y
1003:b
1001:∧
999:x
995:y
991:x
987:c
985:∨
983:b
979:a
975:y
971:c
967:b
965:∧
963:a
959:x
955:a
951:c
947:c
945:,
943:b
941:,
939:a
932:a
928:a
926:∨
924:b
920:a
916:c
914:∨
912:b
908:a
904:c
900:b
898:∧
896:a
892:c
888:b
886:∧
884:c
880:c
858:)
855:c
852:=
849:a
846:(
838:)
833:)
830:b
824:c
821:=
818:b
812:a
809:(
801:)
798:b
792:c
789:=
786:b
780:a
777:(
769:)
766:a
760:c
757:(
752:(
737:G
733:c
729:b
725:a
720:G
709:k
705:k
701:k
679:5
676:N
669:b
665:a
661:x
657:b
653:b
649:x
645:x
641:b
637:a
633:x
626:b
622:x
618:a
614:a
610:b
606:x
602:b
598:a
594:x
590:5
587:N
555:.
552:b
548:a
546:∨
544:x
542:(
540:=
538:b
534:b
530:x
526:x
522:)
520:b
518:∧
516:a
512:x
507:5
504:N
475:.
472:b
468:x
464:b
460:a
456:b
452:x
448:b
444:a
442:(
429:b
425:a
419:a
413:b
409:b
405:a
399:b
395:a
391:a
385:b
381:a
374:.
372:)
370:b
366:x
362:a
358:b
354:x
350:a
348:(
343:b
339:a
325:b
321:x
317:a
313:b
309:x
305:a
299:b
295:a
286:b
282:x
278:a
274:b
270:x
266:a
260:b
256:a
248:x
244:x
205:)
203:b
199:x
197:(
193:a
184:b
180:a
175:a
171:x
167:b
120:b
116:a
112:x
103:b
99:x
95:a
91:b
87:x
83:a
77:b
73:a
42:.
20:.
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