Characteristic Set

	Characteristic Set Characteristic set may refer to * The characteristic set of an algebraic matroid * The characteristic set of a linear matroid * Wu's method of characteristic set {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Algebraic Matroid In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence. Definition Given a field extension ''L''/''K'', Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of ''L'' over ''K''. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. For every finite set ''S'' of elements of ''L'', the algebraically independent subsets of ''S'' satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set ''T'' of elements is the intersection of ''L'' with the field ''K'' 'T''Oxley (1992) p.216 A matroid that can be generated in this way is called ''algebraic'' or ''algebraically representable''.Oxley (1992) p.218 No good characterization of algebraic matroids is k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Linear Matroid In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra. A linear matroid is a matroid that has a representation, and an ''F''-linear matroid (for a field ''F'') is a matroid that has a representation using a vector space over ''F''. Matroid representation theory studies the existence of representations and the properties of linear matroids. Definitions A (finite) matroid (E,\mathcal) is defined by a finite set E (the elements of the matroid) and a non-empty family \mathcal of the subsets of E, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]