By Manfred Stern (auth.)
ISBN-10: 3663124789
ISBN-13: 9783663124788
ISBN-10: 3663124797
ISBN-13: 9783663124795
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Sample text
That they show up in so many places and disguises suggests that they are worthwhile objects of study. Matroid theory is today a highly developed branch of combinatorics. Besides the book by WELSH [1976] we also refer to AIGNER [1975, 1976, 1979], CRAPO-ROTA [1970 a] and TUTTE [1971]. The analogy with vector spaces leads to the following notions. The rank function of a matroid is a function r: 2 5 --+ ~ defined by r(A) =max (lXI :X~ A, X E F) (As; 5). The rank r(M) of the matroid M is the rank of the set 5.
Obtained a deep extension of Dilworth's Covering Theorem: s. KUNG [1985, 1986 a, 1987); we remark that these papers also contain interesting arithmetical results for (finite) semimodular lattices. GANTER-RIVAL [1975] extended equality (++) above to the modular case; this follows from their stronger result that every semimodular lattice of finite length satisfies l(M(L)) S l(J(L)). We note that equality (++) also holds in certain nonmodular semimodular lattices of finite length. For instance, any geometric lattice Lis atomistic as well dually atomistic and thus l(M(L)) • 1 l(J(L)) holds in this case.
2. Lattices (of finite length) in which the KORP holds for meet decompositions have been characterized by CRAWLEY [1961); for upper semimodular lattices (of finite length) this has been done by DILWORTH [1941 b) using the concept of local modularity (s. also CRAWLEY-DILWORTH [1973]). on the other hand, (upper) semimodular lattices of finite length in which the KORP holds for join-decompositions have been characterized by FAIGLE [1980 b] using the concept of strongness (s. also REUTER [1989]). This latter case will be treated in more detail in section 27 where we also clarify the relationship of the results obtained by Dilworth and Faigle.
Semimodular Lattices by Manfred Stern (auth.)
by Christopher
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