Numerically regular hereditary classes of combinatorial geometries
✍ Scribed by Joseph P. S. Kung
- Publisher
- Springer
- Year
- 1986
- Tongue
- English
- Weight
- 914 KB
- Volume
- 21
- Category
- Article
- ISSN
- 0046-5755
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✦ Synopsis
A hereditary class 24 ° of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in ~ is extremal if no proper extension of G of the same rank is in Yr. The size function h(n) of ~ is defined by h(n)-max{[G[:GEgf and rank(G)-n}, where [G] is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in ,;/t ~, [ G] = h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n) -h(n -1): 1 < n < oo} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.
ra(X ) + r~(Y) = rG(X w Y) + rG(X c~ Y).