منابع مشابه
Lifts of matroid representations over partial fields
There exist several theorems which state that when a matroid is representable over distinct fields F1, . . . ,Fk, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle’s characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a m...
متن کاملConfinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P′ ⊆ P. We say that B confines M to P′ if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P′-matrix. We show that, under some conditions on the partial fields, on M , and on B, verifying whether B confines M to P′ amounts to a finite check. A corollary of t...
متن کاملOriented Lagrangian Orthogonal Matroid Representations
Several attempts have been made to extend the theory of matroids (here referred to as ordinary or classical matroids) to theories of more general objects, in particular the Coxeter matroids of Borovik, Gelfand and White ([7], first introduced as WP-matroids in [10]), and the ∆-matroids and (equivalent but for notation) symmetric matroids of Bouchet (see, for example, [8]). The special cases of ...
متن کاملMatroid Representations and Free Arrangements
We show that Terao's Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this. 1. Free arrangements The present study continues an investigation of the connection between algebraic and combinator...
متن کاملPartial Fields and Matroid Representation
A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a, b ∈ P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the class of matroids representable over a partial field. The matroids representabl...
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ژورنال
عنوان ژورنال: European Journal of Combinatorics
سال: 1984
ISSN: 0195-6698
DOI: 10.1016/s0195-6698(84)80041-4