744:
732:
963:
If the characteristic set of a matroid is infinite, it contains zero; and if it contains zero then it contains all but finitely many primes. Hence the only possible characteristic sets are finite sets not containing zero, and cofinite sets containing zero. Indeed, all such sets do occur.
1147:
of mechanical linkages formed by rigid bars connected at their ends by flexible hinges. A linkage of this type may be described as a graph, with an edge for each bar and a vertex for each hinge, and for one-dimensional linkages the rigidity matroids are exactly the graphic matroids.
890:-linear matroids can be characterized by a finite set of forbidden minors, similar to the characterizations described above for the binary and regular matroids. As of 2012, it has been proven only for fields of four or fewer elements. For infinite fields (such as the field of the
1255:. Every linear matroid is algebraic, and for fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist algebraic matroids that are not linear.
759:
is one of the smallest matroids that is unrepresentable over all fields. If a matroid is linear, it may be representable over some but not all fields. For instance, the nine-element rank-three matroid defined by the
707:, with one column per matroid element and with a set of elements being independent in the matroid if and only if the corresponding set of matrix columns is linearly independent. The
207:, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one independent set
480:
444:
405:
135:
277:
349:
185:
1176:
1007:
862:
820:
612:
1238:
641:
1211:
1091:
1071:
1047:
1027:
693:
661:
580:
560:
540:
520:
500:
369:
325:
297:
245:
225:
205:
158:
699:
if and only if the underlying matroid is simple (having no two-element dependent sets). Matroid representations may also be described more concretely using
1361:
52:; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of
1569:
1622:
1731:
Oxley, James; Semple, Charles; Vertigan, Dirk; Whittle, Geoff (2002), "Infinite antichains of matroids with characteristic set {
830:
are the matroids that can be represented over all fields; they can be characterized as the matroids that have none of
299:
to form a larger independent set. One of the key motivating examples in the formulation of matroids was the notion of
1708:
1676:
1311:
1283:
1577:
1528:
1486:
1144:
17:
1418:
1913:, Contemporary Mathematics, vol. 197, Providence, RI: American Mathematical Society, pp. 171–311,
872:
of the Fano plane as minors. Alternatively, a matroid is regular if and only if it can be represented by a
1523:
449:
413:
374:
104:
1664:
250:
873:
330:
166:
942:
716:
1159:
2005:
978:
833:
791:
1668:
1658:
1252:
1073:-dimensional vector space. The field of representation must be large enough for there to exist
903:
1602:
712:
643:
is linearly independent. A matroid with a representation is called a linear matroid, and if
585:
1986:
1963:
1928:
1892:
1801:
1766:
1643:
1598:
1551:
1509:
1470:
1444:
1382:
1216:
1125:
879:
761:
748:
700:
49:
1809:
1718:
1686:
960:
of prime numbers there exists a matroid whose characteristic set is the given finite set.
617:
48:
relation is the same as that of a given matroid. Matroid representations are analogous to
8:
1278:, Oxford Graduate Texts in Mathematics, vol. 3, Oxford University Press, p. 8,
300:
72:
45:
1196:
1076:
1056:
1032:
1012:
678:
646:
565:
545:
525:
505:
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354:
310:
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230:
210:
190:
143:
1748:
1704:
1672:
1542:
1500:
1307:
1279:
1244:
1102:
1193:, are linear over every sufficiently large field. More specifically, a gammoid with
952:
there exist infinitely many matroids whose characteristic set is the singleton set {
1951:
1914:
1880:
1805:
1789:
1752:
1744:
1714:
1682:
1631:
1586:
1537:
1495:
1430:
1370:
1153:
1140:
1121:
1050:
1124:
by defining a set of edges to be independent if and only if it does not contain a
756:
736:
1982:
1959:
1924:
1906:
1888:
1797:
1762:
1639:
1594:
1547:
1505:
1466:
1458:
1440:
1378:
1248:
1117:
973:
827:
786:
769:
90:
studies the existence of representations and the properties of linear matroids.
1919:
1793:
1635:
1484:
Bixby, Robert E. (1979), "On Reid's characterization of the ternary matroids",
1190:
775:
161:
53:
1942:
Lindström, Bernt (1973), "On the vector representations of induced matroids",
1999:
1699:
Ingleton, A.W. (1971), "Representation of matroids", in Welsh, D.J.A. (ed.),
1663:, Encyclopedia of Mathematics and its Applications, vol. 29, Cambridge:
1299:
823:
671:-linear matroid. Thus, the linear matroids are exactly the matroids that are
1590:
1186:
1049:
of the elements. Uniform matroids may be represented by sets of vectors in
946:
907:
869:
779:
708:
304:
79:
41:
1979:
Combinatorial
Mathematics and its Applications (Proc. Conf., Oxford, 1969)
1955:
1701:
Combinatorial mathematics and its applications. Proceedings, Oxford, 1969
1435:
1093:
vectors in general position in this vector space, so uniform matroids are
1620:
Vámos, P. (1978), "The missing axiom of matroid theory is lost forever",
1356:
1271:
1149:
891:
765:
720:
672:
1105:, the direct sums of the uniform matroids, as the direct sum of any two
675:
to the matroids defined from sets or multisets of vectors. The function
1757:
1565:
957:
865:
743:
138:
1463:
Actes du Congrès
International des Mathématiciens (Nice, 1970), Tome 3
1148:
Higher-dimensional rigidity matroids may be defined using matrices of
696:
1884:
1374:
1029:
elements, and its independent sets consist of all subsets of up to
1182:
99:
33:
731:
1213:
elements may be represented over every field that has at least
782:
921:
is the minimal subfield of its algebraic closure over which
1730:
1909:(1996), "Some matroids from discrete applied geometry",
1423:
Journal of
Research of the National Bureau of Standards
1977:
Ingleton, A. W. (1971), "Representation of matroids",
1780:
Kahn, Jeff (1982), "Characteristic sets of matroids",
1570:"The excluded minors for GF(4)-representable matroids"
726:
715:
of submatrices of this matrix, or equivalently by the
1219:
1199:
1162:
1079:
1059:
1035:
1015:
981:
836:
794:
785:; they are exactly the matroids that do not have the
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620:
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377:
357:
333:
313:
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233:
213:
193:
169:
146:
107:
1564:
1359:(1958), "A homotopy theorem for matroids. I, II",
1232:
1205:
1181:Like uniform matroids and partition matroids, the
1170:
1085:
1065:
1041:
1021:
1001:
945:of the fields over which it is linear. For every
856:
814:
778:are the matroids that can be represented over the
687:
655:
635:
606:
574:
554:
534:
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494:
474:
438:
399:
363:
343:
319:
291:
271:
239:
219:
199:
179:
152:
129:
1362:Transactions of the American Mathematical Society
351:is the family of linearly independent subsets of
78:) is a matroid that has a representation using a
1997:
1247:are matroids defined from sets of elements of a
1128:. Every graphic matroid is regular, and thus is
1944:The Bulletin of the London Mathematical Society
868:(a binary matroid with seven elements), or the
755:Not every matroid is linear; the eight-element
63:is a matroid that has a representation, and an
967:
160:(the elements of the matroid) and a non-empty
1871:Graver, Jack E. (1991), "Rigidity matroids",
1526:(1979), "Matroid representation over GF(3)",
1465:, Paris: Gauthier-Villars, pp. 229–233,
1461:(1971), "Combinatorial theory, old and new",
941:of a linear matroid is defined as the set of
751:, linear over the reals but not the rationals
1981:, London: Academic Press, pp. 149–167,
1120:is the matroid defined from the edges of an
327:is a finite set or multiset of vectors, and
1782:Journal of the London Mathematical Society
1623:Journal of the London Mathematical Society
1306:, Courier Dover Publications, p. 10,
29:Vectors with given pattern of independence
1941:
1918:
1756:
1541:
1499:
1434:
1412:
1410:
1351:
1349:
1347:
1164:
1097:-linear for all but finitely many fields
446:is any matroid, then a representation of
1976:
1905:
1698:
1568:; Gerards, A. M. H.; Kapoor, A. (2000),
1152:with a structure similar to that of the
894:) no such characterization is possible.
742:
730:
227:is larger than a second independent set
1522:
1156:of the underlying graph, and hence are
14:
1998:
1870:
1650:
1407:
1344:
897:
1858:
1846:
1840:
1834:
1828:
1822:
1816:
1692:
1656:
1619:
1483:
1416:
1398:
1355:
1338:
1326:
1298:
1270:
932:
1873:SIAM Journal on Discrete Mathematics
1779:
1703:, Academic Press, pp. 149–167,
1457:
1389:
882:states that, for every finite field
711:of a linear matroid is given by the
1516:
1289:. For the rank function, see p. 26.
727:Characterization of linear matroids
24:
1911:Matroid theory (Seattle, WA, 1995)
542:, with the property that a subset
475:{\displaystyle (E,{\mathcal {I}})}
464:
439:{\displaystyle (E,{\mathcal {I}})}
428:
400:{\displaystyle (E,{\mathcal {I}})}
389:
336:
172:
130:{\displaystyle (E,{\mathcal {I}})}
119:
25:
2017:
272:{\displaystyle x\in A\setminus B}
263:
1970:
1935:
1899:
1864:
1852:
1773:
1724:
1613:
1578:Journal of Combinatorial Theory
1558:
1529:Journal of Combinatorial Theory
1487:Journal of Combinatorial Theory
1477:
1451:
1332:
1320:
1292:
1264:
929:can be taken to be of rank 3.
667:then the matroid is called an
630:
624:
594:
582:is independent if and only if
469:
453:
433:
417:
394:
378:
344:{\displaystyle {\mathcal {I}}}
180:{\displaystyle {\mathcal {I}}}
124:
108:
93:
32:In the mathematical theory of
13:
1:
1749:10.1016/S0012-365X(00)00466-0
1258:
663:is a vector space over field
482:may be defined as a function
247:then there exists an element
88:Matroid representation theory
1543:10.1016/0095-8956(79)90055-8
1501:10.1016/0095-8956(79)90056-X
1171:{\displaystyle \mathbb {R} }
7:
1109:-linear matroids is itself
1101:. The same is true for the
1002:{\displaystyle U{}_{n}^{r}}
968:Related classes of matroids
857:{\displaystyle U{}_{4}^{2}}
815:{\displaystyle U{}_{4}^{2}}
739:, not linear over any field
10:
2022:
1665:Cambridge University Press
764:is representable over the
1657:White, Neil, ed. (1987),
874:totally unimodular matrix
1794:10.1112/jlms/s2-26.2.207
1660:Combinatorial geometries
1636:10.1112/jlms/s2-18.3.403
1185:, matroids representing
1132:-linear for every field
723:of subsets of vectors.
1920:10.1090/conm/197/02540
1591:10.1006/jctb.2000.1963
1419:"Lectures on matroids"
1253:algebraic independence
1234:
1207:
1172:
1087:
1067:
1043:
1023:
1003:
904:algebraic number field
858:
816:
752:
740:
689:
657:
637:
608:
607:{\displaystyle f|_{A}}
576:
556:
536:
516:
496:
476:
440:
401:
365:
345:
321:
293:
273:
241:
221:
201:
181:
154:
131:
38:matroid representation
1436:10.6028/jres.069b.001
1417:Tutte, W. T. (1965),
1235:
1233:{\displaystyle 2^{n}}
1208:
1173:
1088:
1068:
1044:
1024:
1004:
859:
817:
746:
734:
690:
658:
638:
609:
577:
557:
537:
517:
497:
477:
441:
402:
366:
346:
322:
294:
279:that can be added to
274:
242:
222:
202:
182:
155:
132:
50:group representations
1737:Discrete Mathematics
1251:using the notion of
1217:
1197:
1160:
1077:
1057:
1033:
1013:
979:
925:can be represented:
834:
826:. The unimodular or
792:
762:Perles configuration
749:Perles configuration
679:
647:
636:{\displaystyle f(A)}
618:
586:
566:
546:
526:
506:
486:
450:
414:
375:
355:
331:
311:
283:
251:
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211:
191:
167:
144:
105:
1956:10.1112/blms/5.1.85
1341:, pp. 170–172, 196.
998:
913:there is a matroid
898:Field of definition
853:
811:
410:More generally, if
301:linear independence
46:linear independence
1245:algebraic matroids
1230:
1203:
1168:
1145:degrees of freedom
1103:partition matroids
1083:
1063:
1039:
1019:
999:
985:
939:characteristic set
933:Characteristic set
854:
840:
812:
798:
753:
741:
685:
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522:to a vector space
512:
492:
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397:
361:
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317:
289:
269:
237:
217:
197:
187:of the subsets of
177:
150:
127:
1784:, Second Series,
1626:, Second Series,
1404:White (1987) p.12
1206:{\displaystyle n}
1141:rigidity matroids
1086:{\displaystyle n}
1066:{\displaystyle r}
1042:{\displaystyle r}
1022:{\displaystyle n}
956:}, and for every
880:Rota's conjecture
768:but not over the
688:{\displaystyle f}
656:{\displaystyle V}
614:is injective and
575:{\displaystyle E}
555:{\displaystyle A}
535:{\displaystyle V}
515:{\displaystyle E}
495:{\displaystyle f}
364:{\displaystyle E}
320:{\displaystyle E}
292:{\displaystyle B}
240:{\displaystyle B}
220:{\displaystyle A}
200:{\displaystyle E}
153:{\displaystyle E}
16:(Redirected from
2013:
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1601:, archived from
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1300:Welsh, D. J. A.
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1272:Oxley, James G.
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1249:field extension
1224:
1220:
1218:
1215:
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1195:
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1191:directed graphs
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1118:graphic matroid
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776:Binary matroids
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1585:(2): 247–299,
1557:
1536:(2): 159–173,
1524:Seymour, P. D.
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1494:(2): 174–204,
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407:is a matroid.
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69:linear matroid
61:linear matroid
54:linear algebra
28:
18:Linear matroid
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1710:0-12-743350-3
1706:
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1678:0-521-33339-3
1674:
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1616:
1608:on 2010-09-24
1604:
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1579:
1571:
1567:
1566:Geelen, J. F.
1561:
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1323:
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1313:9780486474397
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1285:9780199202508
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1246:
1241:
1225:
1221:
1200:
1192:
1188:
1184:
1179:
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1151:
1146:
1143:describe the
1142:
1137:
1135:
1131:
1127:
1123:
1119:
1114:
1112:
1108:
1104:
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1096:
1080:
1060:
1052:
1036:
1016:
994:
989:
982:
975:
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951:
948:
944:
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807:
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795:
788:
784:
781:
777:
773:
771:
767:
763:
758:
757:Vámos matroid
750:
745:
738:
737:Vámos matroid
733:
724:
722:
718:
714:
710:
709:rank function
706:
703:over a field
702:
698:
682:
674:
670:
666:
650:
627:
621:
599:
589:
569:
549:
529:
509:
489:
459:
456:
423:
420:
408:
384:
381:
358:
314:
306:
302:
286:
266:
260:
257:
254:
234:
214:
194:
163:
147:
140:
114:
111:
101:
91:
89:
85:
81:
77:
74:
70:
66:
62:
57:
55:
51:
47:
43:
39:
35:
27:
19:
1978:
1972:
1947:
1943:
1937:
1910:
1901:
1876:
1872:
1866:
1859:Oxley (2006)
1854:
1847:Oxley (2006)
1842:
1835:Oxley (2006)
1830:
1823:Oxley (2006)
1818:
1785:
1781:
1775:
1740:
1736:
1732:
1726:
1700:
1694:
1659:
1652:
1627:
1621:
1615:
1603:the original
1582:
1581:, Series B,
1576:
1560:
1533:
1532:, Series B,
1527:
1518:
1491:
1490:, Series B,
1485:
1479:
1462:
1453:
1426:
1422:
1400:
1391:
1366:
1360:
1357:Tutte, W. T.
1339:Oxley (2006)
1334:
1327:Oxley (2006)
1322:
1303:
1294:
1275:
1266:
1242:
1187:reachability
1180:
1150:real numbers
1138:
1133:
1129:
1115:
1110:
1106:
1098:
1094:
971:
962:
953:
949:
947:prime number
938:
936:
926:
922:
918:
914:
910:
908:finite field
901:
892:real numbers
887:
883:
878:
870:dual matroid
780:finite field
774:
766:real numbers
754:
704:
668:
664:
409:
305:vector space
97:
87:
83:
80:vector space
75:
68:
64:
60:
58:
37:
31:
26:
1758:10092/13245
1369:: 144–174,
721:linear span
713:matrix rank
98:A (finite)
94:Definitions
1810:0468.05020
1719:0222.05025
1687:0626.00007
1667:, p.
1259:References
1240:elements.
958:finite set
917:for which
906:and every
902:For every
866:Fano plane
697:one-to-one
673:isomorphic
502:that maps
139:finite set
1950:: 85–90,
1861:, p. 100.
1849:, p. 228.
1837:, p. 226.
1825:, p. 225.
1178:-linear.
1113:-linear.
717:dimension
264:∖
258:∈
2000:Category
1429:: 1–47,
1329:, p. 12.
1302:(2010),
1274:(2006),
1183:gammoids
701:matrices
695:will be
34:matroids
1987:0278974
1964:0335313
1929:1411692
1893:1105942
1802:0675165
1767:1874763
1644:0518224
1599:1769191
1552:0532586
1510:0532587
1471:0505646
1445:0179781
1383:0101526
719:of the
371:, then
100:matroid
71:(for a
42:vectors
1985:
1962:
1927:
1891:
1808:
1800:
1765:
1717:
1707:
1685:
1675:
1642:
1597:
1550:
1508:
1469:
1443:
1381:
1310:
1282:
1053:in an
886:, the
864:, the
162:family
44:whose
1606:(PDF)
1573:(PDF)
1126:cycle
824:minor
822:as a
783:GF(2)
307:: if
82:over
73:field
1735:}",
1705:ISBN
1673:ISBN
1308:ISBN
1280:ISBN
1243:The
1139:The
1009:has
937:The
747:The
735:The
36:, a
1952:doi
1915:doi
1881:doi
1806:Zbl
1790:doi
1753:hdl
1745:doi
1741:242
1715:Zbl
1683:Zbl
1632:doi
1587:doi
1538:doi
1496:doi
1431:doi
1427:69B
1371:doi
1189:in
562:of
2002::
1983:MR
1960:MR
1958:,
1946:,
1925:MR
1923:,
1889:MR
1887:,
1875:,
1804:,
1798:MR
1796:,
1786:26
1763:MR
1761:,
1751:,
1739:,
1713:,
1681:,
1671:,
1669:18
1640:MR
1638:,
1628:18
1595:MR
1593:,
1583:79
1575:,
1548:MR
1546:,
1534:26
1506:MR
1504:,
1492:26
1467:MR
1441:MR
1439:,
1425:,
1421:,
1409:^
1379:MR
1377:,
1367:88
1365:,
1346:^
1136:.
1116:A
972:A
876:.
772:.
86:.
59:A
56:.
1990:.
1967:.
1954::
1948:5
1932:.
1917::
1896:.
1883::
1877:4
1813:.
1792::
1770:.
1755::
1747::
1733:p
1647:.
1634::
1610:.
1589::
1555:.
1540::
1513:.
1498::
1474:.
1448:.
1433::
1386:.
1373::
1317:.
1226:n
1222:2
1201:n
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1130:F
1111:F
1107:F
1099:F
1095:F
1081:n
1061:r
1037:r
1017:n
995:r
990:n
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954:p
950:p
927:M
923:M
919:F
915:M
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888:F
884:F
850:2
845:4
838:U
808:2
803:4
796:U
705:F
683:f
669:F
665:F
651:V
631:)
628:A
625:(
622:f
600:A
595:|
590:f
570:E
550:A
530:V
510:E
490:f
470:)
465:I
460:,
457:E
454:(
434:)
429:I
424:,
421:E
418:(
395:)
390:I
385:,
382:E
379:(
359:E
337:I
315:E
287:B
267:B
261:A
255:x
235:B
215:A
195:E
173:I
148:E
125:)
120:I
115:,
112:E
109:(
84:F
76:F
67:-
65:F
20:)
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