Matroid

Definition

In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.

Related concepts

A. M. H. Gerards Abstract simplicial complex Advances in Mathematics Aequationes Mathematicae Algebra Algebraic independence Algebraic matroid American Journal of Mathematics Antimatroid Arrangement of hyperplanes Atom (order theory)Axiomatic system Banach space Bartel Leendert van der Waerden Basis (linear algebra)Basis of a matroid Biased graph Bicircular matroid Binary matroid Binghamton University Bipartite graph Brooklyn College Cambridge University Press Characteristic polynomial Chromatic polynomial City University of New York Closure operator Coding theory Combinatorial optimization Combinatorics Complete graph Convex hull Covering relation Coxeter matroid Cryptomorphism Cycle graph Cyclomatic number Disjoint union Dominic Welsh Dowling geometry Dual graph Dual matroid Empty set Encyclopedia of Mathematics Ernst Steinitz Eugene Lawler European Journal of Combinatorics European Mathematical Society Extension field Family of sets Fano plane Field (mathematics)Field theory (mathematics)Finite field Finite geometry Finite set Florida Atlantic University Free matroid GF(2)Gain graph Gammoid Geoff Whittle Geometric lattice Geometry George J. Minty Gian-Carlo Rota Graph theory Graphic matroid Greedoid Greedy algorithm Hassler Whitney Hazel Perfect Henry Crapo (mathematician)Hilbert space If and only if Independence Theory in Combinatorics Independence system Indicator vector Integer Internet Archive Jack Edmonds James Oxley Jim Geelen Journal of Combinatorial Theory Journal of Mathematics and Mechanics Journal of the London Mathematical Society Kuratowski's theorem Laman graph Linear Algebra and Its Applications Linear algebra Linear independence Logical matrix Louisiana State University László Lovász Macaulay2 Maple (software)Matching (graph theory)Mathematical Association of America Mathematical logic Mathematics Magazine Matrix (mathematics)Matroid, Inc.Matroid girth Matroid intersection Matroid lattice Matroid minor Matroid partitioning Matroid polytope Matroid rank Matroid representation Maximal element Maximal independent set Maximum matching Meteoroid Metroid Minimal element Model theory Monotonic function Multigraph Multiset Möbius function (combinatorics)NP-complete Nancy Neudauer Natural number Network theory Oriented matroid Partially ordered set Partition matroid Partition of a set Paul Seymour (mathematician)Paving matroid Planar graph Polymatroid Polynomial time Power set Pregeometry (model theory)Projective geometry Pseudoforest Real number Regular matroid Richard Rado Rigidity matroid Robertson–Seymour theorem Rota's conjecture SageMath Saunders Mac Lane Signed graph Simple cycle Singleton (mathematics)Sperner family Steinitz exchange lemma Stirling number Structural rigidity Submodular function Subset Takeo Nakasawa Thomas H. Brylawski Topology Transactions of the American Mathematical Society Transcendence degree Transversal (combinatorics)Tree (graph theory)Tutte homotopy theorem Tutte polynomial Uniform matroid University of Washington Vector space Vámos matroid W. T. Tutte Weighted matroid

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