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| In [[combinatorics]], a branch of [[mathematics]], a '''matroid''' {{IPAc-en|ˈ|m|eɪ|t|r|ɔɪ|d}} or '''independence structure''' is a structure that captures and generalizes the notion of [[linear independence]] in [[vector space]]s.
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| There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.
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| Matroid theory borrows extensively from the terminology of [[linear algebra]] and [[graph theory]], largely because it is the abstraction of various notions of central importance in these fields.
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| ==Definition==<!-- [[Hereditary property (matroid)]] redirects to this section title-->
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| There are many equivalent ([[Cryptomorphism|cryptomorphic]]) ways to define a (finite) matroid.<ref name="oxley">A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Bryzlawski's appendix in White (1986) pp.298–302 for a list of equivalent axiom systems.</ref>
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| ===Independent sets===
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| In terms of independence, a finite matroid <math>M</math> is a pair <math>(E,\mathcal{I})</math>, where <math>E</math> is a [[finite set]] (called the '''ground set''') and <math>\mathcal{I}</math> is a [[family of sets|family]] of [[subset]]s of <math>E</math> (called the '''independent sets''') with the following properties:<ref name="w7-9">{{harvtxt|Welsh|1976}}, Section 1.2, "Axiom Systems for a Matroid", pp. 7–9.</ref>
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| # The [[empty set]] is independent, i.e., <math>\emptyset\in\mathcal{I}</math>. Alternatively, at least one subset of <math>E</math> is independent, i.e., <math>\mathcal{I}\neq\emptyset</math>.
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| # Every subset of an independent set is independent, i.e., for each <math>A'\subset A\subset E</math>, if <math>A\in\mathcal{I}</math> then <math>A'\in\mathcal{I}</math>. This is sometimes called the '''hereditary property'''.
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| # If <math>A</math> and <math>B</math> are two independent sets of <math>\mathcal{I}</math> and <math>A</math> has more elements than<math>B</math>, then there exists an element in <math>A</math> that when added to <math>B</math> gives a larger independent set than <math>B</math>. This is sometimes called the '''augmentation property''' or the '''independent set exchange property'''.
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| The first two properties define a combinatorial structure known as an [[independence system]].
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| ===Bases and circuits===
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| A subset of the ground set <math>E</math> that is not independent is called '''dependent'''. A maximal independent set—that is, an independent set which becomes dependent on adding any element of <math>E</math>—is called a '''basis''' for the matroid. A '''circuit''' in a matroid <math>M</math> is a minimal dependent subset of <math>E</math>—that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of [[graphic matroid]]s are cycles in the corresponding graphs.<ref name="w7-9"/>
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| The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely: a set is independent if and only if it is not dependent, if and only if it is a subset of a basis, and if and only if it does not contain a circuit. The collection of dependent sets, or of bases, or of circuits each has simple properties that may be taken as axioms for a matroid. For instance, one may define a matroid <math>M</math> to be a pair <math>(E,\mathcal{B})</math>, where <math>E</math> is a finite set as before and <math>\mathcal{B}</math> is a collection of subsets of <math>E</math>, called "bases", with the following properties:<ref name="w7-9"/>
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| # <math>\mathcal{B}</math> is nonempty.
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| # If <math>A</math> and <math>B</math> are distinct members of <math>\mathcal{B}</math> and <math>a\in A\setminus B</math>, then there exists an element <math>b\in B\setminus A</math> such that <math>A\setminus\{a\}\cup\{b\}\in\mathcal{B}</math>. (Here the backslash symbol stands for the [[Complement (set theory)|difference of sets]]. This property is called the '''basis exchange property'''.)
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| It follows from the basis exchange property that no member of <math>\mathcal{B}</math> can be a proper subset of another.
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| ===Rank functions===
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| It is a basic result of matroid theory, directly analogous to a similar theorem of [[basis (linear algebra)|bases in linear algebra]], that any two bases of a matroid <math>M</math> have the same number of elements. This number is called the '''[[matroid rank|rank]]''' of <math>M</math>. If <math>M</math> is a matroid on <math>E</math>, and <math>A</math> is a subset of <math>E</math>, then a matroid on <math>A</math> can be defined by considering a subset of <math>A</math> to be independent if and only if it is independent in <math>M</math>. This allows us to talk about '''submatroids''' and about the rank of any subset of <math>E</math>. The rank of a subset ''A'' is given by the '''[[matroid rank|rank function]]''' ''r''(''A'') of the matroid, which has the following properties:<ref name="w7-9"/>
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| *The value of the rank function is always a non-negative [[integer]].
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| *For any subset <math>A</math> of <math>E</math>, <math>r(A) \le |A|</math>.
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| *For any two subsets <math>A</math> and <math>B</math> of <math>E</math>, <math>r(A\cup B)+r(A\cap B)\le r(A)+r(B)</math>. That is, the rank is a [[submodular function]].
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| *For any set <math>A</math> and element <math>x</math>, <math>r(A)\le r(A\cup\{x\})\le r(A)+1</math>. From the first of these two inequalities it follows more generally that, if <math>A\subset B\subset E</math>, then <math>r(A)\leq r(B)\leq r(E)</math>. That is, the rank is a [[monotonic function]].
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| These properties can be used as one of the alternative definitions of a finite matroid: if <math>(E,r)</math> satisfies these properties, then the independent sets of a matroid over <math>E</math> can be defined as those subsets <math>A</math> of <math>E</math> with <math>r(A)=|A|</math>.
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| The difference <math>|A|-r(A)</math> is called the '''nullity''' or '''corank''' of the subset <math>A</math>. It is the minimum number of elements that must be removed from <math>A</math> to obtain an independent set. The nullity of <math>E</math> in <math>M</math> is called the nullity or corank of <math>M</math>.
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| ===Closure operators===
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| Let <math>M</math> be a matroid on a finite set <math>E</math>, with rank function <math>r</math> as above. The '''closure''' <math>\operatorname{cl}(A)</math> of a subset <math>A</math> of <math>E</math> is the set
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| :<math>\operatorname{cl}(A) = \Bigl\{x\in E\mid r(A)=r\bigl(A\cup\{x\}\bigr)\Bigr\}</math>.
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| This defines a [[closure operator]] <math>\operatorname{cl}: \mathcal{P}(E)\to \mathcal{P}(E)</math> where <math>\mathcal{P}</math> denotes the [[power set]], with the following properties:
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| * For all subsets <math>X</math> of <math>E</math>, <math>X\subseteq \operatorname{cl}(X)</math>.
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| * For all subsets <math>X</math> of <math>E</math>, <math>\operatorname{cl}(X)= \operatorname{cl}(\operatorname{cl}(X))</math>.
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| * For all subsets <math>X</math> and <math>Y</math> of <math>E</math> with <math>X\subseteq Y</math>, <math>\operatorname{cl}(X)\subseteq \operatorname{cl}(Y)</math>.
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| * For all elements <math>a</math>, and <math>b</math> of <math>E</math> and all subsets <math>Y</math> of <math>E</math>, if <math>a\in\operatorname{cl}(Y\cup \{b\}) \setminus \operatorname{cl}(Y)</math> then <math>b\in\operatorname{cl}(Y\cup \{a\}) \setminus \operatorname{cl}(Y)</math>.
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| The first three of these properties are the defining properties of a closure operator. The fourth is sometimes called the '''Mac Lane–Steinitz exchange property'''. These properties may be taken as another definition of matroid: every function <math>\operatorname{cl}: \mathcal{P}(E)\to \mathcal{P}(E)</math> that obeys these properties determines a matroid.<ref name="w7-9"/>
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| ===Flats===
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| A set whose closure equals itself is said to be '''closed''', or a '''flat''' or '''subspace''' of the matroid.<ref>{{harvtxt|Welsh|1976}}, Section 1.8, "Closed sets = Flats = Subspaces", pp. 21–22.</ref> A set is closed if it is [[maximal element|maximal]] for its rank, meaning that the addition of any other element to the set would increase the rank. The closed sets of a matroid are characterized by a covering partition property:
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| * The whole point set <math>E</math> is closed.
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| * If <math>S</math> and <math>T</math> are flats, then <math>S\cap T</math> is a flat.
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| * If <math>S</math> is a flat, then the flats <math>T</math> that [[Covering relation|cover]] <math>S</math> (meaning that <math>T</math> properly contains <math>S</math> but there is no flat <math>U</math> between <math>S</math> and <math>T</math>), partition the elements of <math>E\setminus S</math>.
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| The class <math>\mathcal{L}(M)</math> of all flats, [[partially ordered set|partially ordered]] by set inclusion, forms a [[matroid lattice]].
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| Conversely, every matroid lattice <math>L</math> forms a matroid over its set <math>E</math> of [[Atom (order theory)|atoms]] under the following closure operator: for a set <math>S</math> of atoms with join <math>\vee S</math>,
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| :<math>\operatorname{cl}(S) = \{ x\in E\mid x\le\vee S \}</math>.
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| The flats of this matroid correspond one-for-one with the elements of the lattice; the flat corresponding to lattice element <math>y</math> is the set
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| :<math>\{ x\in E\mid x\le y\}</math>.
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| Thus, the lattice of flats of this matroid is naturally isomorphic to <math>L</math>.
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| ===Hyperplanes===
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| In a matroid of rank <math>r</math>, a flat of rank <math>r-1</math> is called a hyperplane. These are the maximal proper flats; that is, the only superset of a hyperplane that is also a flat is the set <math>E</math> of all the elements of the matroid. Hyperplanes are also called '''coatoms''' or '''copoints'''. An equivalent definition: A coatom is a subset of ''E'' that does not span ''M'', but such that adding any other element to it does make a spanning set.<ref name="w38-39">{{harvtxt|Welsh|1976}}, Section 2.2, "The Hyperplanes of a Matroid", pp. 38–39.</ref>
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| The family <math>\mathcal{H}</math> of hyperplanes of a matroid has the following properties, which may be taken as yet another axiomatization of matroids:<ref name="w38-39"/>
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| *There do not exist distinct sets <math>X</math> and <math>Y</math> in <math>\mathcal{H}</math> with <math>X\subset Y</math>. That is, the hyperplanes form a [[Sperner family]].
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| *For every <math>x\in E</math> and <math>Y,Z\in\mathcal{H}</math> with <math>x\notin Y\cup Z</math>, there exists <math>X\in\mathcal{H}</math> with <math>(Y\cap Z)\cup\{x\}\subseteq X</math>.
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| == Examples ==
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| ===Uniform matroids===
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| Let ''E'' be a finite set and ''k'' a [[natural number]]. One may define a matroid on ''E'' by taking every ''k''-element subset of ''E'' to be a basis. This is known as the [[uniform matroid]] of rank ''k''. A uniform matroid with rank ''k'' and with ''n'' elements is denoted <math>U_{k,n}</math>. All uniform matroids of rank at least 2 are simple. The uniform matroid of rank 2 on ''n'' points is called the ''n''-'''point line'''. A matroid is uniform if and only if it has no circuits of size less than the one plus the rank of the matroid. The direct sums of uniform matroids are called [[partition matroid]]s.
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| In the uniform matroid <math>U_{0,n}</math>, every element is a loop (an element that does not belong to any independent set), and in the uniform matroid <math>U_{n,n}</math>, every element is a coloop (an element that belongs to all bases). The direct sum of matroids of these two types is a partition matroid in which every element is a loop or a coloop; it is called a '''discrete matroid'''. An equivalent definition of a discrete matroid is a matroid in which every proper, non-empty subset of the ground set ''E'' is a separator.
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| ===Matroids from linear algebra===
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| [[File:fano plane.svg|thumb|The Fano matroid, derived from the [[Fano plane]]. It is [[GF(2)]]-linear but not real-linear.]]
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| [[File:Vamos matroid.svg|thumb|The [[Vámos matroid]], not linear over any field]]
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| Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces. There are two ways to present the matroids defined in this way:
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| * If ''E'' is any finite subset of a [[vector space]] ''V'', then we can define a matroid ''M'' on ''E'' by taking the independent sets of ''M'' to be the [[linear independence|linearly independent]] subsets of ''E''. The validity of the independent set axioms for this matroid follows from the [[Steinitz exchange lemma]]. If ''M'' is a matroid that can be defined in this way, we say the set ''E'' '''[[matroid representation|represents]]''' ''M''. Matroids of this kind are called '''vector matroids'''. An important example of a matroid defined in this way is the Fano matroid, a rank-three matroid derived from the [[Fano plane]], a [[finite geometry]] with seven points (the seven elements of the matroid) and seven lines (the nontrivial flats of the matroid). It is a linear matroid whose elements may be described as the seven nonzero points in a three-dimensional vector space over the [[finite field]] [[GF(2)]]. However, it is not possible to provide a similar representation for the Fano matroid using the [[real number]]s in place of GF(2).
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| * A [[matrix (mathematics)|matrix]] ''A'' with entries in a [[field (mathematics)|field]] gives rise to a matroid ''M'' on its set of columns. The dependent sets of columns in the matroid are those that are linearly dependent as vectors. This matroid is called the '''column matroid''' of ''A'', and ''A'' is said to '''represent''' ''M''. For instance, the Fano matroid can be represented in this way as a 3 × 7 [[Logical matrix|(0,1)-matrix]]. Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation. (There is one technical difference: a column matroid can have distinct elements that are the same vector, but a vector matroid as defined above cannot. Usually this difference is insignificant and can be ignored, but by letting ''E'' be a [[multiset]] of vectors one brings the two definitions into complete agreement.)
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| A matroid that is equivalent to a vector matroid, although it may be presented differently, is called '''representable''' or '''linear'''. If ''M'' is equivalent to a vector matroid over a [[field (mathematics)|field]] ''F'', then we say ''M'' is '''representable over''' ''F'' ; in particular, ''M'' is '''real-representable''' if it is representable over the real numbers. For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroid theory is to characterize the matroids that may be represented over a given field ''F''; [[Rota's conjecture]] describes a possible characterization for every [[finite field]]. The main results so far are characterizations of [[binary matroid]]s (those representable over GF(2)) due to Tutte (1950s), of ternary matroids (representable over the 3-element field) due to Reid and Bixby, and separately to Seymour (1970s), and of quaternary matroids (representable over the 4-element field) due to Geelen, Gerards, and Kapoor (2000). This is very much an open area.
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| A [[regular matroid]] is a matroid that is representable over all possible fields. The [[Vámos matroid]] is the simplest example of a matroid that is not representable over any field.
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| ===Matroids from graph theory===
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| A second original source for the theory of matroids is [[graph theory]].
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| Every finite graph (or [[multigraph]]) ''G'' gives rise to a matroid ''M''(''G'') as follows: take as ''E'' the set of all edges in ''G'' and consider a set of edges independent if and only if it is a [[tree (graph theory)|forest]]; that is, if it does not contain a [[simple cycle]]. Matroids derived in this way are '''[[graphic matroid]]s'''. Not every matroid is graphic, but all matroids on three elements are graphic.<ref name=Ox13/> Every graphic matroid is regular.
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| Other matroids on graphs were discovered subsequently:
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| *The [[bicircular matroid]] of a graph is defined by calling a set of edges independent if every connected subset contains at most one cycle.
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| *In any directed or undirected graph ''G'' let ''E'' and ''F'' be two distinguished sets of vertices. In the set ''E'', define a subset ''U'' to be independent if there are |''U''| vertex-disjoint paths from ''F'' onto ''U''. This defines a matroid on ''E'' called a '''[[gammoid]]''':<ref name=Ox115/> a '''strict gammoid''' is one for which the set ''E'' is the whole vertex set of ''G''.<ref name=Ox100>{{harvnb|Oxley|1992|p=100}}</ref>
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| *In a [[bipartite graph]] ''G'' = (''U'',''V'',''E''), one may form a matroid in which the elements are vertices on one side ''U'' of the bipartition, and the independent subsets are sets of endpoints of [[Matching (graph theory)|matchings]] of the graph. This is called a '''transversal matroid''',<ref name=Ox4648>{{harvnb|Oxley|1992|pp=46–48}}</ref><ref name=Wh877297>{{White|1987|pp=72–97}}</ref> and it is a special case of a gammoid.<ref name=Ox115>{{harvnb|Oxley|1992|pp=115}}</ref> The transversal matroids are the [[dual matroid]]s to the strict gammoids.<ref name=Ox100/>
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| *Graphic matroids have been generalized to matroids from [[signed graph]]s, [[gain graph]]s, and [[biased graph]]s. A graph ''G'' with a distinguished linear class '''''B''''' of cycles, known as a "biased graph" (''G'','''''B'''''), has two matroids, known as the '''frame matroid''' and the '''lift matroid''' of the biased graph. If every cycle belongs to the distinguished class, these matroids coincide with the cycle matroid of ''G''. If no cycle is distinguished, the frame matroid is the bicircular matroid of ''G''. A signed graph, whose edges are labeled by signs, and a gain graph, which is a graph whose edges are labeled orientably from a group, each give rise to a biased graph and therefore have frame and lift matroids.
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| *The [[Laman graph]]s form the bases of the two-dimensional [[rigidity matroid]], a matroid defined in the theory of [[structural rigidity]].
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| ===Matroids from field extensions===
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| A third original source of matroid theory is [[field theory (mathematics)|field theory]].
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| An [[extension field|extension]] of a field gives rise to a matroid. Suppose ''F'' and ''K'' are fields with ''K'' containing ''F''. Let ''E'' be any finite subset of ''K''. Define a subset ''S'' of ''E'' to be [[algebraic independence|algebraically independent]] if the extension field ''F''(''S'') has [[transcendence degree]] equal to |''S''|.<ref name=Ox215>{{harvnb|Oxley|1992|p=215}}</ref>
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| A matroid that is equivalent to a matroid of this kind is called an '''algebraic matroid'''.<ref name=Ox216>{{harvnb|Oxley|1992|p=216}}</ref> The problem of characterizing algebraic matroids is extremely difficult; little is known about it. The [[Vámos matroid]] provides an example of a matroid that is not algebraic.
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| == Basic constructions ==
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| There are some standard ways to make new matroids out of old ones.
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| ===Duality===
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| If ''M'' is a finite matroid, we can define the '''orthogonal''' or '''[[dual matroid]]''' ''M''* by taking the same underlying set and calling a set a ''basis'' in ''M''* if and only if its complement is a basis in ''M''. It is not difficult to verify that ''M''* is a matroid and that the dual of ''M''* is ''M''.<ref name=Whi8632>{{harvnb|White|1986|p=32}}</ref>
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| The dual can be described equally well in terms of other ways to define a matroid. For instance:
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| * A set is independent in ''M''* if and only if its complement spans ''M''.
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| * A set is a circuit of ''M''* if and only if its complement is a coatom in ''M''.
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| * The rank function of the dual is r*(S) = |S|- r(E) + r(E\S).
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| According to a matroid version of [[Kuratowski's theorem]], the dual of a graphic matroid ''M'' is a graphic matroid if and only if ''M'' is the matroid of a [[planar graph]]. In this case, the dual of ''M'' is the matroid of the [[dual graph]] of ''G''.<ref name=Whi86105>{{harvnb|White|1986|p=105}}</ref> The dual of a vector matroid representable over a particular field ''F'' is also representable over ''F''. The dual of a transversal matroid is a strict gammoid and vice versa.
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| ===Minors===
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| If ''M'' is a matroid with element set ''E'', and ''S'' is a subset of ''E'', the '''restriction''' of ''M'' to ''S'', written ''M'' |''S'', is the matroid on the set ''S'' whose independent sets are the independent sets of ''M'' that are contained in ''S''. Its circuits are the circuits of ''M'' that are contained in ''S'' and its rank function is that of ''M'' restricted to subsets of ''S''. In linear algebra, this corresponds to restricting to the subspace generated by the vectors in ''S''. Equivalently if ''T'' = ''M''−''S'' this may be termed the '''deletion''' of ''T'', written ''M''\''T'' or ''M''−''T''. The submatroids of ''M'' are precisely the results of a sequence of deletions: the order is irrelevant.<ref name=Whi86131>{{harvnb|White|1986|p=131}}</ref><ref name=Whi86224>{{harvnb|White|1986|p=224}}</ref>
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| The dual operation of restriction is contraction.<ref name=Whi866139>{{harvnb|White|1986|p=139}}</ref> If ''T'' is a subset of ''E'', the '''contraction''' of ''M'' by ''T'', written ''M''/''T'', is the matroid on the underlying set ''E − T'' whose rank function is <math>r'(A) = r(A \cup T) - r(T).</math><ref name=Whi86140>{{harvnb|White|1986|p=140}}</ref> In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in ''T'', together with the images of the vectors in ''E - T''.
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| A matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations is called a [[matroid minor|minor]] of ''M''.<ref name=Whi86224/><ref name=Whi86150>{{harvnb|White|1986|p=150}}</ref> We say ''M'' '''contains''' ''N'' '''as a minor'''. Many important families of matroids may be characterized by the [[minimal element|minor-minimal]] matroids that do not belong to the family; these are called '''forbidden''' or '''excluded minors'''.<ref name=Whi861467>{{harvnb|White|1986|pp=146–147}}</ref>
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| ===Sums and unions===
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| Let ''M'' be a matroid with an underlying set of elements ''E'', and let ''N'' be another matroid on an underlying set ''F''.
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| The '''direct sum''' of matroids ''M'' and ''N'' is the matroid whose underlying set is the [[disjoint union]] of ''E'' and ''F'', and whose independent sets are the disjoint unions of an independent set of ''M'' with an independent set of ''N''.
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| The '''union''' of ''M'' and ''N'' is the matroid whose underlying set is the union (not the disjoint union) of ''E'' and ''F'', and whose independent sets are those subsets which are the union of an independent set in ''M'' and one in ''N''. Usually the term "union" is applied when ''E'' = ''F'', but that assumption is not essential. If ''E'' and ''F'' are disjoint, the union is the direct sum.
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| == Additional terminology ==
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| Let ''M'' be a matroid with an underlying set of elements ''E''.
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| * ''E'' may be called the '''ground set''' of ''M''. Its elements may be called the '''points''' of ''M''.
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| * A subset of ''E'' '''spans''' ''M'' if its closure is ''E''. A set is said to '''span''' a closed set ''K'' if its closure is ''K''.
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| * An element that forms a single-element circuit of ''M'' is called a '''loop'''. Equivalently, an element is a loop if it belongs to no basis.<ref name=Ox13/><ref name=Wh86130>{{harvnb|White|1986|p=130}}</ref>
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| * An element that belongs to no circuit is called a '''coloop''' or '''isthmus'''. Equivalently, an element is a coloop if it belongs to every basis. Loop and coloops are mutually dual.<ref name=Wh86130/>
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| * If a two-element set {''f, g''} is a circuit of ''M'', then ''f'' and ''g'' are '''parallel''' in ''M''.<ref name=Ox13/>
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| * A matroid is called '''simple''' if it has no circuits consisting of 1 or 2 elements. That is, it has no loops and no parallel elements. The term '''combinatorial geometry''' is also used.<ref name=Ox13>{{harvnb|Oxley|1992|p=13}}</ref> A simple matroid obtained from another matroid ''M'' by deleting all loops and deleting one element from each 2-element circuit until no 2-element circuits remain is called a '''simplification''' of ''M''.<ref name=Ox52>{{harvnb|Oxley|1992|p=52}}</ref> A matroid is '''co-simple''' if its dual matroid is simple.<ref name=Ox347>{{harvnb|Oxley|1992|p=347}}</ref>
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| * A union of circuits is sometimes called a '''cycle''' of ''M''. A cycle is therefore the complement of a flat of the dual matroid. (This usage conflicts with the common meaning of "cycle" in graph theory.)
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| * A '''separator''' of ''M'' is a subset ''S'' of ''E'' such that <math>r(S) + r(E-S) = r(M)</math>. A '''proper''' or '''non-trivial separator''' is a separator that is neither ''E'' nor the empty set.<ref name=Ox128>{{harvnb|Oxley|1992|p=128}}</ref> An '''irreducible separator''' is a separator that contains no other non-empty separator. The irreducible separators partition the ground set ''E''.
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| * A matroid which cannot be written as the direct sum of two nonempty matroids, or equivalently which has no proper separators, is called '''connected''' or '''irreducible'''. A matroid is connected if and only if its dual is connected.<ref name=Wh86110>{{harvnb|White|1986|p=110}}</ref>
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| * A maximal irreducible submatroid of ''M'' is called a '''component''' of ''M''. A component is the restriction of ''M'' to an irreducible separator, and contrariwise, the restriction of ''M'' to an irreducible separator is a component. A separator is a union of components.<ref name=Ox128/>
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| * A matroid ''M'' is called a '''frame matroid''' if it, or a matroid that contains it, has a basis such that all the points of ''M'' are contained in the lines that join pairs of basis elements.<ref>{{cite journal | zbl=0797.05027 | last=Zaslavsky | first=Thomas | title=Frame matroids and biased graphs | journal=Eur. J. Comb. | volume=15 | number=3 | pages=303–307 | year=1994 | issn=0195-6698 | url=http://www.sciencedirect.com/science/article/pii/S0195669884710341 | doi=10.1006/eujc.1994.1034}}</ref>
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| * A matroid is called a [[paving matroid]] if all of its circuits have size at least equal to its rank.<ref name=Ox26>{{harvnb|Oxley|1992|p=26}}</ref>
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| ==Algorithms==
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| ===Greedy algorithm===
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| A [[weighted matroid]] is a matroid together with a function from its elements to the nonnegative [[real number]]s. The weight of a subset of elements is defined to be the sum of the weights of the elements in the subset. The [[greedy algorithm]] can be used to find a maximum-weight basis of the matroid, by starting from the empty set and repeatedly adding one element at a time, at each step choosing a maximum-weight element among the elements whose addition would preserve the independence of the augmented set.<ref name=Ox63>{{harvnb|Oxley|1992|p=63}}</ref> This algorithm does not need to know anything about the details of the matroid's definition, as long as it has access to the matroid through an [[matroid oracle|independence oracle]], a subroutine for testing whether a set is independent.
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| This optimization algorithm may be used to characterize matroids: if a family ''F'' of sets, closed under taking subsets, has the property that, no matter how the sets are weighted, the greedy algorithm finds a maximum-weight set in the family, then ''F'' must be the family of independent sets of a matroid.<ref name=Ox64>{{harvnb|Oxley|1992|p=64}}</ref>
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| The notion of matroid has been generalized to allow for other types of sets on which a greedy algorithm give optimal solutions; see [[greedoid]] and [[matroid embedding]] for more information.
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| ===Matroid partitioning===
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| The [[matroid partitioning]] problem is to partition the elements of a matroid into as few independent sets as possible, and the matroid packing problem is to find as many disjoint spanning sets as possible. Both can be solved in polynomial time, and can be generalized to the problem of computing the rank or finding an independent set in a matroid sum.
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| ===Matroid intersection===
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| The [[matroid intersection|intersection]] of two or more matroids is the family of sets that are simultaneously independent in each of the matroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids can be found in [[polynomial time]],and provides a solution to many other important combinatorial optimization problems. For instance, [[maximum matching]] in [[bipartite graph]]s can be expressed as a problem of intersecting two [[partition matroid]]s. However, finding the largest set in an intersection of three or more matroids is [[NP-complete]].
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| ===Matroid software===
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| Two standalone systems for calculations with matroids are Kingan's [http://userhome.brooklyn.cuny.edu/skingan/software.html Oid] and Hlineny's [http://www.fi.muni.cz/~hlineny/MACEK/ Macek]. Both of them are open sourced packages. "Oid" is an interactive, extensible software system for experimenting with matroids. "Macek" is a specialized software system with tools and routines for reasonably efficient combinatorial computations with representable matroids.
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| SAGE, the open source mathematics software system, will contain a matroid package from release 5.12.
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| ==Polynomial invariants==
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| There are two especially significant polynomials associated to a finite matroid ''M'' on the ground set ''E''. Each is a '''matroid invariant''', which means that isomorphic matroids have the same polynomial.
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| ===Characteristic polynomial===
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| The '''characteristic polynomial''' of ''M'' (which is sometimes called the '''chromatic polynomial''',<ref name=Wh87127/> although it does not count colorings), is defined to be
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| :<math>p_M(\lambda) := \sum_{S \subseteq E} (-1)^{|S|}\lambda^{r(M)-r(S)},</math>
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| or equivalently (as long as the empty set is closed in ''M'') as
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| :<math>p_M(\lambda) := \sum_A \mu(\emptyset,A) \lambda^{r(M)-r(A)} \ ,</math>
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| where μ denotes the [[Möbius function (combinatorics)|Möbius function]] of the [[geometric lattice]] of the matroid.<ref name=Wh87120>{{harvnb|White|1987|p=120}}</ref>
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| When ''M'' is the cycle matroid ''M''(''G'') of a graph ''G'', the characteristic polynomial is a slight transformation of the [[chromatic polynomial]], which is given by χ<sub>''G''</sub> (λ) = λ<sup>c</sup>''p''<sub>''M''(''G'')</sub> (λ), where ''c'' is the number of connected components of ''G''.
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| When ''M'' is the bond matroid ''M''*(''G'') of a graph ''G'', the characteristic polynomial equals the [[Tutte polynomial#Flow polynomial|flow polynomial]] of ''G''.
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| When ''M'' is the matroid ''M''(''A'') of an [[Arrangement of hyperplanes|arrangement]] ''A'' of linear hyperplanes in '''R'''<sup>''n''</sup> (or ''F''<sup>''n''</sup> where ''F'' is any field), the characteristic polynomial of the arrangement is given by ''p''<sub>''A''</sub> (λ) = λ<sup>''n''−''r''(''M'')</sup>''p''<sub>''M''(''A'')</sub> (λ).
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| ====Beta invariant====
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| The '''beta invariant''' of a matroid, introduced by Crapo (1967), may be expressed in terms of the characteristic polynomial ''p'' as an evaluation of the derivative<ref name=Wh87123>{{harvnb|White|1987|p=123}}</ref>
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| :<math> \beta(M) = (-1)^{r(M)-1} p_M'(1) \ </math>
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| or directly as<ref name=Wh87124>{{harvnb|White|1987|p=124}}</ref>
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| :<math> \beta(M) = (-1)^{r(M)} \sum_{X \subseteq E} (-1)^{|X|} r(X) \ . </math>
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| The beta invariant is non-negative, and is zero if and only if ''M'' is disconnected, or empty, or a loop. Otherwise it depends only on the lattice of flats of ''M''. If ''M'' has no loops and coloops then β(''M'') = β(''M''<sup>∗</sup>).<ref name=Wh87124/>
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| ===Tutte polynomial===
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| The '''[[Tutte polynomial]]''' of a matroid, ''T''<sub>''M''</sub> (''x'',''y''), generalizes the characteristic polynomial to two variables. This gives it more combinatorial interpretations, and also gives it the duality property
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| :<math>T_{M^*}(x,y) = T_M(y,x),</math>
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| which implies a number of dualities between properties of ''M'' and properties of ''M'' *. One definition of the Tutte polynomial is
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| :<math>T_M(x,y) = \sum_{S\subseteq E} (x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.</math>
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| This expresses the Tutte polynomial as an evaluation of the '''corank-nullity''' or '''rank generating polynomial''',<ref name=Wh87126>{{harvnb|White|1987|p=126}}</ref>
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| :<math>R_M(u,v) = \sum_{S\subseteq E} u^{r(M)-r(S)}v^{|S|-r(S)}.</math>
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| From this definition it is easy to see that the characteristic polynomial is, up to a simple factor, an evaluation of ''T''<sub>''M''</sub>, specifically,
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| :<math>p_M(\lambda) = (-1)^{r(M)} T_M(1-\lambda,0). </math>
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| Another definition is in terms of internal and external activities and a sum over bases, reflecting the fact that ''T''(1,1) is the number of bases.<ref name=Wh92188>{{harvnb|White|1992|p=188}}</ref> This, which sums over fewer subsets but has more complicated terms, was Tutte's original definition.
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| There is a further definition in terms of recursion by deletion and contraction.<ref name=Wh86260>{{harvnb|White|1986|p=260}}</ref> The deletion-contraction identity is
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| :<math>F(M) = F(M-e)+F(M/e)</math> when <math>e</math> is neither a loop nor a coloop.
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| An invariant of matroids (i.e., a function that takes the same value on isomorphic matroids) satisfying this recursion and the multiplicative condition
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| :<math>F(M\oplus M') = F(M) F(M')</math>
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| is said to be a '''Tutte-Grothendieck invariant'''.<ref name=Wh87126/> The Tutte polynomial is the most general such invariant; that is, the Tutte polynomial is a Tutte-Grothendieck invariant and every such invariant is an evaluation of the Tutte polynomial.<ref name=Wh87127>{{harvnb|White|1987|p=127}}</ref>
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| The [[Tutte polynomial]] ''T''<sub>''G''</sub> of a graph is the Tutte polynomial ''T''<sub>''M''(''G'')</sub> of its cycle matroid.
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| == Infinite matroids ==
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| <!-- [[Infinite matroid]] redirects here. -->
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| The theory of infinite matroids is much more complicated than that of finite matroids and forms a subject of its own. For a long time, one of the difficulties has been that there were many reasonable and useful definitions, none of which appeared to capture all the important aspects of finite matroid theory. For instance, it seemed to be hard to have bases, circuits, and duality together in one notion of infinite matroids.
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| The simplest definition of an infinite matroid is to require ''finite rank''; that is, the rank of ''E'' is finite. This theory is similar to that of finite matroids except for the failure of duality due to the fact that the dual of an infinite matroid of finite rank does not have finite rank. Finite-rank matroids include any subsets of finite-dimensional vector spaces and of [[Field (mathematics)|field extensions]] of finite [[transcendence degree]].
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| The next simplest infinite generalization is finitary matroids. A matroid is '''finitary''' if it has the property that
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| :<math>x \in cl(Y) \Leftrightarrow (\exists Y' \subseteq Y) Y' \text{ is finite and } x \in cl(Y').</math>
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| Equivalently, every dependent set contains a finite dependent set.
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| Examples are linear dependence of arbitrary subsets of infinite-dimensional [[vector space]]s (but not infinite dependencies as in [[Hilbert space|Hilbert]] and [[Banach space]]s), and algebraic dependence in arbitrary subsets of field extensions of possibly infinite transcendence degree. Again, the class of finitary matroid is not self-dual, because the dual of a finitary matroid is not finitary.
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| Finitary infinite matroids are studied in [[model theory]], a branch of [[mathematical logic]] with strong ties to [[algebra]].
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| In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroids and generalizes their duality. Many notions of infinite matroids were defined in response to this challenge, but the question remained open. One of the approaches examined by D.A. Higgs became known as ''B-matroids'' and was studied by Higgs, Oxley and others in the 1960s and 1970s. According to a recent result by Bruhn, Diestel, Kriesell, Pendavingh and Wollan ([[#CITEREFBruhnDiestelKriesellWollan2010|2010]]), it solves the problem: Arriving at the same notion independently, they provided four different systems of axioms – in terms of independence, bases, circuits, closure and rank. The duality of B-matroids generalizes dualities that can be observed in infinite graphs.
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| ==History==
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| Matroid theory was introduced by {{harvs|last=Whitney|first=Hassler|authorlink=Hassler Whitney|year=1935|txt}}. It was also independently discovered by [[Takeo Nakasawa]], whose work was forgotten for many years {{harv|Nishimura|Kuroda|2009}}.
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| In his seminal paper, Whitney provided two axioms for independence, and defined any structure adhering to these axioms to be "matroids".
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| (Although it was perhaps implied, he did not include an axiom requiring at least one subset to be independent.)
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| His key observation was that these axioms provide an abstraction of "independence" that is common to both graphs and matrices.
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| Because of this, many of the terms used in matroid theory resemble the terms for their analogous concepts in [[linear algebra]] or [[graph theory]].
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| Almost immediately after Whitney first wrote about matroids, an important article was written by {{harvs|last=Mac Lane|first=Saunders|authorlink=Saunders Mac Lane|year=1936|txt}} on the relation of matroids to projective geometry. A year later, {{harvs|last=van der Waerden|first=B. L.|authorlink=Bartel Leendert van der Waerden|year=1937|txt}} noted similarities between algebraic and linear dependence in his classic textbook on Modern Algebra.
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| In the 1940s [[Richard Rado]] developed further theory under the name "independence systems" with an eye towards [[Transversal (combinatorics)|transversal theory]], where his name for the subject is still sometimes used.
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| In the 1950s [[W. T. Tutte]] became the foremost figure in matroid theory, a position he retained for many years. His contributions were plentiful, including the characterization of [[binary matroid|binary]], [[regular matroid|regular]], and [[graphic matroid|graphic]] matroids by [[Matroid minor|excluded minors]]; the regular-matroid representability theorem; the theory of chain groups and their matroids; and the tools he used to prove many of his results, the "Path theorem" and "[[Tutte homotopy theorem|Homotopy theorem]]" (see, e.g., {{harvnb|Tutte|1965}}), which are so complex that later theorists have gone to great trouble to eliminate the necessity of using them in proofs. (A fine example is [[A. M. H. Gerards]]' short proof ([[#CITEREFGerards1989|1989]]) of Tutte's characterization of regular matroids.)
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| {{harvs|first=Henry|last=Crapo|year=1969|txt}} and {{harvs|first=Thomas|last=Brylawski|year=1972|txt}} generalized to matroids Tutte's "dichromate", a graphic polynomial now known as the [[Tutte polynomial]] (named by Crapo). Their work has recently (especially in the 2000s) been followed by a flood of papers—though not as many as on the Tutte polynomial of a graph.
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| In 1976 [[Dominic Welsh]] published the first comprehensive book on matroid theory.
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| [[Paul Seymour (mathematician)|Paul Seymour]]'s decomposition theorem for regular matroids ([[#CITEREFSeymour1980|1980]]) was the most significant and influential work of the late 1970s and the 1980s.
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| Another fundamental contribution, by {{harvtxt|Kahn|Kung|1982}}, showed why projective geometries and Dowling geometries play such an important role in matroid theory.
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| By this time there were many other important contributors, but one should not omit to mention [[Geoff Whittle]]'s extension to ternary matroids of Tutte's characterization of binary matroids that are representable over the rationals {{harv|Whittle|1995}}, perhaps the biggest single contribution of the 1990s. In the current period (since around 2000) the Matroid Minors Project of [[Jim Geelen]], Gerards, Whittle, and others, which attempts to duplicate for matroids that are representable over a finite field the success of the Robertson–Seymour Graph Minors Project (see [[Robertson–Seymour theorem]]), has produced substantial advances in the structure theory of matroids. Many others have also contributed to that part of matroid theory, which (in the first and second decades of the 21st century) is flourishing.
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| ==Researchers==
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| Mathematicians who pioneered the study of matroids include [[Takeo Nakasawa]],{{sfnp|Nishimura|Kuroda|2009}} [[Saunders Mac Lane]], [[Richard Rado]], [[W. T. Tutte]], [[Bartel Leendert van der Waerden|B. L. van der Waerden]], and [[Hassler Whitney]].
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| Other major contributors include [[Jack Edmonds]], [[Jim Geelen]], [[Eugene Lawler]], [[László Lovász]], [[Gian-Carlo Rota]], [[Paul Seymour (mathematician)|P. D. Seymour]], and [[Dominic Welsh]].
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| There is an on-line list of [http://userhome.brooklyn.cuny.edu/skingan/matroids/people.html current researchers].
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| ==See also==
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| * [[Antimatroid]]
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| * [[Coxeter matroid]]
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| * [[Oriented matroid]]
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| * [[Pregeometry (model theory)]]
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| ==Notes==
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| {{reflist|colwidth=30em}}
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| == References ==
| |
| *{{citation |last1=Bruhn |first1=Henning |last2=Diestel |first2=Reinhard |last3=Kriesell |first3=Matthias |last4=Pendavingh |first4=Rudi |last5=Wollan |first5=Paul |title=Axioms for infinite matroids |year=2010 |arxiv=1003.3919 }}.
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| *{{citation|last1=Bryant|first1=Victor|last2=Perfect|first2=Hazel|year=1980|title=Independence Theory in Combinatorics|publisher=Chapman and Hall|location=London and New York|isbn=0-412-22430-5}}.
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| *{{citation|last=Brylawski|first=Thomas H.|year=1972|title=A decomposition for combinatorial geometries|journal=Transactions of the American Mathematical Society|volume=171|pages=235–282|doi=10.2307/1996381|publisher=American Mathematical Society|jstor=1996381}}.
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| *{{citation|last=Crapo|first=Henry H.|year=1969|title=The Tutte polynomial|journal=Aequationes Mathematicae|volume=3|issue=3|pages=211–229|doi=10.1007/BF01817442}}.
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| *{{citation|last1=Crapo|first=Henry H.|last2=Rota|first2=Gian-Carlo|author2-link=Gian-Carlo Rota|year=1970|title=On the Foundations of Combinatorial Theory: Combinatorial Geometries|publisher=M.I.T. Press|location=Cambridge, Mass.|isbn=978-0-262-53016-3|mr=0290980}}.
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| *{{citation|last1=Geelen|first1=Jim|last2=Gerards|first2=A. M. H.|last3=Whittle|first3=Geoff|year=2007|contribution=Towards a matroid-minor structure theory|editor=Grimmett, Geoffrey (ed.) et al|title=Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh|series=Oxford Lecture Series in Mathematics and its Applications|volume=34|pages=72–82|publisher=Oxford University Press|publication-place=Oxford}}.
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| *{{citation|last=Gerards|first=A. M. H.|year=1989|title=A short proof of Tutte's characterization of totally unimodular matrices|journal=[[Linear Algebra and its Applications]]|volume=114/115|pages=207–212|doi=10.1016/0024-3795(89)90461-8}}.
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| *{{citation|last1=Kahn|first1=Jeff|last2=Kung|first2=Joseph P. S.|year=1982|title=Varieties of combinatorial geometries|journal=Transactions of the American Mathematical Society|volume=271|pages=485–499|doi=10.2307/1998894|issue=2|publisher=American Mathematical Society|jstor=1998894}}.
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| *{{citation|last1=Kingan|first1=Robert|last2=Kingan|first2=Sandra | year=2005|contribution=A software system for matroids|title=Graphs and Discovery|series=DIMACS Series in Discrete Mathematics and Theoretical Computer Science|pages=287–296}}.
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| *{{citation|editor-last=Kung|editor-first=Joseph P. S.|title=A Source Book in Matroid Theory|publisher=Birkhäuser|isbn=0-8176-3173-9|location=Boston|year=1986|mr=0890330}}.
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| *{{citation|last=Mac Lane|first=Saunders|authorlink=Saunders Mac Lane|year=1936|title=Some interpretations of abstract linear dependence in terms of projective geometry|journal=American Journal of Mathematics|volume=58|pages=236–240|doi=10.2307/2371070|issue=1|publisher=The Johns Hopkins University Press|jstor=2371070}}.
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| *{{citation|mr=2516551|zbl=1163.01001|title=A lost mathematician, Takeo Nakasawa. The forgotten father of matroid theory|editor-first=Hirokazu |editor-last=Nishimura |editor2-first=Susumu |editor2-last=Kuroda|publisher= Birkhäuser Verlag|place= Basel|year= 2009|isbn= 978-3-7643-8572-9|url=http://www.springerlink.com/content/978-3-7643-8572-9}}.
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| *{{citation|last=Oxley|first=James | authorlink = James Oxley|year=1992|title=Matroid Theory|publisher=Oxford University Press|location=Oxford|isbn=0-19-853563-5|mr=1207587|zbl=0784.05002}}.
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| *{{citation|last=Recski|first=András|year=1989|title=Matroid Theory and its Applications in Electric Network Theory and in Statics|publisher=Springer-Verlag and Akademiai Kiado|location=Berlin and Budapest|isbn=3-540-15285-7|mr=1027839}}.
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| *{{eom|id=M/m062870|first=A.A.|last= Sapozhenko}}
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| *{{citation|last=Seymour|first=Paul D.|authorlink=Paul Seymour (mathematician)|year=1980|title=Decomposition of regular matroids|journal=Journal of Combinatorial Theory, Series B|volume=28|issue=3|pages=305–359|doi=10.1016/0095-8956(80)90075-1|zbl=0443.05027}}.
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| *{{citation|last=Truemper|first=Klaus|title=Matroid Decomposition|publisher=Academic Press|location=Boston|year=1992|isbn=0-12-701225-7|url=http://www.emis.de/monographs/md/index.html|mr=1170126}}.
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| *{{citation|last=Tutte|first=W. T.|authorlink=W. T. Tutte|year=1959|title=Matroids and graphs|journal=Transactions of the American Mathematical Society|volume=90|pages=527–552|doi=10.2307/1993185|issue=3|publisher=American Mathematical Society|mr=0101527|jstor=1993185}}.
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| *{{citation|last=Tutte|first=W. T.|authorlink=W. T. Tutte|year=1965|title=Lectures on matroids|journal=Journal of Research of the National Bureau of Standards (U.S.A.), Sect. B|volume=69|pages=1–47}}.
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| *{{citation | zbl=0231.05027 | last=Tutte | first=W.T. | authorlink=W. T. Tutte | title=Introduction to the theory of matroids | series=Modern Analytic and Computational Methods in Science and Mathematics | volume=37 | location=New York | publisher=American Elsevier Publishing Company | year=1971 }}.
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| *{{citation|last=Vámos|first=Peter|year=1978|title=The missing axiom of matroid theory is lost forever|journal=Journal of the London Mathematical Society|volume=18|pages=403–408|doi=10.1112/jlms/s2-18.3.403|issue=3}}.
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| *{{citation|last=van der Waerden|first=B. L.|authorlink=Bartel Leendert van der Waerden|year=1937|title=Moderne Algebra}}.
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| *{{citation|last=Welsh|first=D. J. A.|year=1976|title=Matroid Theory|publisher=Academic Press|isbn=0-12-744050-X|zbl=0343.05002|series=L.M.S. Monographs | volume=8}}.
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| *{{citation|editor-last=White|editor-first=Neil|year=1986|title=Theory of Matroids|series=Encyclopedia of Mathematics and its Applications|volume=26|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-30937-0 | zbl=0579.00001 }}.
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| *{{citation | editor-last=White | editor-first=Neil | title=Combinatorial geometries | series=Encyclopedia of Mathematics and its Applications | volume=29 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1987 | isbn=0-521-33339-3 | zbl=0626.00007 }}
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| *{{citation|editor-last=White|editor-first=Neil|year=1992|title=Matroid Applications|series=Encyclopedia of Mathematics and its Applications|volume=40|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-38165-9 | zbl=0742.00052 }}.
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| *{{citation|last=Whitney|first=Hassler|authorlink=Hassler Whitney|year=1935|title=On the abstract properties of linear dependence|journal=American Journal of Mathematics|volume=57|pages=509–533|doi=10.2307/2371182|issue=3|publisher=The Johns Hopkins University Press|mr=1507091|jstor=2371182}}. Reprinted in {{harvtxt|Kung|1986}}, pp. 55–79.
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| *{{citation|last=Whittle|first=Geoff|year=1995|title=A characterization of the matroids representable over ''GF''(3) and the rationals|journal=Journal of Combinatorial Theory Series B|volume=65|issue=2|pages=222–261|url=http://eprints.kfupm.edu.sa/39296/1/39296.pdf|doi=10.1006/jctb.1995.1052}}.
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| == External links ==
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| * {{springer|title=Matroid|id=p/m062870}}
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| * Kingan, Sandra : [http://userhome.brooklyn.cuny.edu/skingan/matroids/ Matroid theory]. A large bibliography of matroid papers, matroid software, and links.
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| * Locke, S. C. : [http://www.math.fau.edu/locke/Greedy.htm Greedy Algorithms].
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| * Pagano, Steven R. : [http://www.math.binghamton.edu/zaslav/Pagano/Matridx.htm Matroids and Signed Graphs].
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| * Mark Hubenthal: [http://www.math.washington.edu/~hubenjm/matroid2.pdf A Brief Look At Matroids] ([[pdf]]) (contain proofs for staments of this article)
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| * James Oxley : [http://www.cs.cornell.edu/courses/CS6820/2012sp/Handouts/oxley-matroids.pdf What is a matroid?]
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| [[Category:Matroid theory| ]]
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| [[Category:Closure operators]]
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| [[Category:Set families]]
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