Semimodular functions and combinatorial geometries
نویسندگان
چکیده
منابع مشابه
Functions of the category of combinatorial geometries and strong maps
As in the case with most mathematical structures, an important question in the theory of combinatorial geometries is to develop constructions for obtaining new geometries from old ones. Several geometric constructions are well-known, e.g. deletion, contraction, truncation, direct sums, etc. . . . and are generally extensions to combinatorial geometries of existing operations on projective geome...
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Let exq(G; n) be the maximum number of points in a rank-n geometry (simple matroid) that is representable over GF (q) and that has no restriction isomorphic to the geometry G. We find exq(G; n) for several infinite families of geometries G, and we show that if G is a binary affine geometry, then lim n→∞ ex2(G; n) 2n − 1 = 0.
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A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class 'S of geometries is a sequence (T„ ) of geometries in ?T with rank Tn = n, and satisfying the universal property: if G is a geometry in 5" of rank n, then G is a subgeometry of T„. A variety of geometries...
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Quasiminimality In this chapter we introduce Zilber’s notion [Zil05] of an abstract quasiminimalexcellent class and prove Theorem 2.23: Lω1,ω-definable quasiminimal-excellent classes satisfying the countable closure condition are categorical in all powers. In the next chapter we expound Zilber’s simplest concrete algebraic example. In Chapter 25, we will place this example in the context of She...
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The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1978
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-1978-0491269-9