Abstract
In his paper [3], Dilworth studied “the manner in which the irreducibles of two decompositions can replace each other” in a modular lattice. Thus this paper anticipated papers on the closely related area of basis exchange properties in matroids or geometric lattices. This relationship is most transparent for modular lattices of finite rank. For such lattices, replacement properties are “local” in the sense that they can be decided by looking only at intervals of the form [a, u a ], where u a is the join all the elements covering a. This follows from the following variant (cf. [5]) of a result in group theory due to Burnside [2] and Frattini [4]
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References
R.A. Brualdi, Comments on bases in dependence structures, Bull. Australian Math. Soc. 1 (1969), 161–167.
W. Burnside, Some properties of groups whose orders are powers of primes, Proc. London Math. Soc. (2) 13 (1913), 6–12.
R.P. Dilworth, Note on the Kurosch-Ore theorem, Bull. Amer. Math. Soc. 52 (1946), 659–663. Reprinted in Chapter 3 of this volume.
G. Frattini, Intorno alla generazione de gruppi di operazioni, Rend. Atti Accad. Lincei (4) 1 (1885), 281–285; 455-457.
K.M. Gragg and J.P.S. Kung, Consistent dually semimodular lattices, preprint.
C. Greene, A multiple exchange property for bases, Proc. Amer. Math. Soc. 39 (1973), 45–50.
C. Greene, Another exchange property for bases, Proc. Amer. Math. Soc. 46 (1974), 155–156.
C. Greene and T.L. Magnanti, Some abstract pivot algorithms, SIAM J. Appl. Math. 29 (1975), 530–539.
J.P.S. Kung, Alternating basis exchanges in matroids, Proc. Amer. Math. Soc. 71 (1978), 355–358.
J.P.S. Kung, Basis exchange properties, in “Theory of Matroids,” N.L. White, ed., Cambridge Univ. Press, Cambridge, 1986, pp. 62–75.
D.R. Woodall, An exchange theorem for bases of matroids, J. Combin. Theory Ser. B 16 (1974), 227–228.
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Kung, J.P.S. (1990). Exchange Properties for Reduced Decompositions in Modular Lattices. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_18
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