Dependencies Among Dependencies in Matroids
Keyword(s):
In 1971, Rota introduced the concept of derived matroids to investigate "dependencies among dependencies" in matroids. In this paper, we study the derived matroid $\delta M$ of an ${\mathbb F}$-representation of a matroid $M$. The matroid $\delta M$ has a naturally associated ${\mathbb F}$-representation, so we can define a sequence $\delta M$, $\delta^2 M$, \dots . The main result classifies such derived sequences of matroids into three types: finite, cyclic, and divergent. For the first two types, we obtain complete characterizations and thereby resolve some of the questions that Longyear posed in 1980 for binary matroids. For the last type, the divergence is estimated by the coranks of the matroids in the derived sequence.
1990 ◽
Vol 49
(2)
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pp. 181-189
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2001 ◽
Vol 10
(5)
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pp. 453-461
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2010 ◽
Vol 133
(1-2)
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pp. 203-225
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1996 ◽
Vol 66
(2)
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pp. 334-360
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2014 ◽
Vol 52
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pp. 1-59
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1989 ◽
Vol 10
(1)
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pp. 17-28
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