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\section*{matroid}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{combinatorics}{}\paragraph*{{Combinatorics}}\label{combinatorics}

[[!include combinatorics - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{modeltheoretic_geometry}{Model-theoretic geometry}\dotfill \pageref*{modeltheoretic_geometry} \linebreak
\noindent\hyperlink{combinatorial_optimization}{Combinatorial optimization}\dotfill \pageref*{combinatorial_optimization} \linebreak
\noindent\hyperlink{mnevs_theorem}{Mnev's theorem}\dotfill \pageref*{mnevs_theorem} \linebreak
\noindent\hyperlink{categories_of_matroids}{Categories of matroids}\dotfill \pageref*{categories_of_matroids} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{matroid}, due to Hassler Whitney, is fundamental to [[combinatorics]], giving several different ways of encoding/defining and presenting a general notion of ``independence'', e.g., [[linearly independent subset|linear independence]] in a [[vector space]], algebraic independence in a [[field extension]], etc.

There is also a similar concept of an [[oriented matroid]]; every oriented matroid has an underlying matroid.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{matroid} on a set $X$ is a [[Moore closure|closure operator]] $cl: P(X) \to P(X)$ satisfying the \emph{exchange axiom}: if $a \in cl(S \cup\{b\}) \cap \neg cl(S)$, then $b \in cl(S \cup\{a\}) \cap \neg cl(S)$.

\end{defn}
Usually when combinatorialists speak of matroids as such, $X$ is taken to be a [[finite set]]. A typical example is $X$ some finite subset of a vector space $V$, taking $cl(S) \coloneqq X \cap Span(S)$ for any $S \subseteq X$.

Under this definition, a subset $S \subseteq X$ is \emph{independent} if there is a strict inclusion $cl(T) \subset cl(S)$ for every strict inclusion $T \subset S$ (this is the same as requiring $x \notin cl(S\setminus \{x\})$ for every $x \in S$). Again under this definition, $S$ is a \emph{basis} if $cl(S) = X$ and $S$ is independent. A \emph{hyperplane} is a closed subset $S$ (meaning $cl(S) = S$) that is maximal among proper closed subsets of $X$. It is possible to axiomatize the notion of matroid by taking bases as the primitive notion, or independent sets as the primitive notion, or hyperplanes as the primitive notion, etc. -- Rota (after Birkhoff) speaks of \emph{cryptomorphism} between these differing definitions. Much of the power and utility of matroid theory comes from this multiplicity of definitions and the possibility of moving seamlessly between them; for example, a matroid structure might be easy to detect from the viewpoint of one definition, but not from another.

\begin{prop}
\label{dim1}\hypertarget{dim1}{}
Any two bases of a matroid $X$ have the same [[cardinality]], provided that one of them is finite.

\end{prop}
The cardinality of such a basis is called of course the \emph{dimension} of the matroid. Clearly then a finite matroid has a well-defined dimension.

\begin{proof}
First, suppose $A$ is an independent set and $B$ is a finite basis, and suppose there are subsets $A_0 \subseteq A, B_0 \subseteq B$ such that $A_0 \cup B_0$ is a basis. We claim that for each $a \in A \setminus A_0$, there exists $b \in B_0$ such that $A_0 \cup \{a\} \cup (B_0 \setminus \{b\})$ is a basis. For, let $C \subseteq B_0$ be of minimum cardinality such that $a \in cl(A_0 \cup C)$; we know $C$ must be inhabited since $a \notin cl(A \setminus \{a\}) \supseteq cl(A_0)$; clearly $C \cap A_0 = \emptyset$. So let $b$ be an element of $C$. Since by minimality of $C$ we have

\begin{displaymath}
a \in cl(A_0 \cup (C \setminus \{b\}) \cup \{b\}) \cap \neg cl(A_0 \cup (C \setminus \{b\})),
\end{displaymath}
it follows from the exchange axiom that $b \in cl(A_0 \cup (C \setminus \{b\}) \cup \{a\})$. Thus $b \in cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\})$, whence

\begin{displaymath}
cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}) = cl(A_0 \cup B_0 \cup \{a\}) = X
\end{displaymath}
so that $D \coloneqq A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}$ ``spans'' $X$. Also $D$ is independent: if $x \in D$ and $x \neq a$, then

\begin{displaymath}
cl(D \setminus \{x\}) \subseteq cl((A_0 \cup B_0) \setminus \{x\})
\end{displaymath}
with neither side containing $x$ since $A_0 \cup B_0$ is independent; whereas if $x = a$ and supposing to the contrary that $a \in cl(D \setminus \{a\}) = cl((A_0 \cup (B_0 \setminus \{b\}))$, we conclude $A_0 \cup (B \setminus \{b\})$ has the same span as $D$. Since $D$ already spans, $b \in cl(A_0 \cup (B_0 \setminus \{b\}))$, again impossible since $A_0 \cup B_0$ is independent. This proves the claim.

Again assuming $A$ independent and $B$ is a finite basis, now we show that $card(A) \leq card(B)$, which will finish the proof. Let $n = card(B)$, and suppose on the contrary that there are distinct elements $a_1, \ldots, a_{n+1} \in A$. Set $A_0 = \emptyset$ and $B_0 = B$. Applying the claim above inductively, we have that $\{a_1, \ldots, a_i\} \cup (B \setminus \{b_1, \ldots, b_i\})$ is a basis for $1 \leq i \leq n$, so in particular $\{a_1, \ldots, a_n\}$ spans $X$. Hence $a_{n+1} \in cl(\{a_1, \ldots, a_{n}\})$, contradicting the independence of $A$.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Vector spaces, algebraic closures, graphs, restrictions, localizations, \ldots{}

\begin{itemize}%
\item A \emph{graphic matroid} is a matroid $M$ derived from a [[simple graph]] by taking the underlying set of $M$ to be the set of edges $E$, and taking as independent sets of $M$ those $S \subseteq E$ such that the edges of $S$ (and their incident vertices) form a [[forest]], i.e., a graph without cycles.


\item A \emph{vector matroid} is a matroid $M$ derived from a a collection of vectors $E$ in a vector space. Independent sets of $M$ are those $S \subseteq E$ such that $S$ is linearly independent. Providing an equivalence to such a vector matroid is the problem of representability over a specified field.


\item An \emph{algebraic matroid} for a field extension $K/F$ is derived from a collection of elements $E \subset K$ and independent sets of $M$ are those such that $F(S)$ has transcendence degree equal to $\mid S \mid$ so all of $S$ are algebraically independent.


\item If $M$ is a (finite) matroid, then the \emph{matroid dual} $M^\ast$ of $M$ has the same underlying set as $M$, and where a basis in $M^\ast$ is precisely the complement of a basis of $M$. It follows that $M^{\ast\ast} \cong M$ (this is even an equality according to the definition).



\end{itemize}
\hypertarget{modeltheoretic_geometry}{}\subsection*{{Model-theoretic geometry}}\label{modeltheoretic_geometry}

Essentially the very same notion arises in model theory, except instead of being called a matroid it is called a ``pregeometry'' or ``geometry'', and in contrast to combinatorialists, model theorists usually mean \emph{infinite} matroids. The notion arises in the study of geometry of strongly minimal sets, with applications to stability theory (part of Shelah's classification theory).

\begin{defn}
\label{}\hypertarget{}{}
A \emph{pregeometry} is a (possibly infinite) matroid (given by a set $X$ equipped with a closure operator $cl$) that is \emph{finitary}: for all $S \subseteq X$, if $x \in cl(S)$ then $x \in cl(S_0)$ for some finite subset $S_0 \subseteq S$. A \textbf{geometry} is a pregeometry such that $cl(\emptyset) = \emptyset$ and $cl(\{x\}) = \{x\}$ for every $x \in X$.

\end{defn}
(See also [[geometric stability theory]].)

The language of independence, spanning, and basis carry over as before. A maximal independent set spans (i.e., is a basis), and maximal independent sets exist according to [[axiom of choice|Zorn's lemma]]. Again we have a notion of dimension by the following proposition.

\begin{prop}
\label{welldefined}\hypertarget{welldefined}{}
In a pregeometry $(X, cl)$, any two bases have the same cardinality.

\end{prop}
\begin{proof}
We already \hyperlink{dim1}{proved this} in the case where the pregeometry has a finite basis. Otherwise, if $A$ is independent and $B$ is an infinite basis, then

\begin{displaymath}
A = A \cap X = A \cap \bigcup_{B_0 \subseteq B\; finite} cl(B_0) = \bigcup_{B_0 \subseteq B\; finite} A \cap cl(B_0)
\end{displaymath}
where the second equality follows from the finitary condition. Since each summand $A \cap cl(B_0)$ has cardinality less than that of $B_0$ by independence of $A$ (noting that $B_0$ is a basis of $cl(B_0)$), the union on the right has cardinality bounded above by $card(B)$. From $card(A) \leq card(B)$ it follows that any two bases have the same cardinality.

\end{proof}
\hypertarget{combinatorial_optimization}{}\subsection*{{Combinatorial optimization}}\label{combinatorial_optimization}

\hypertarget{mnevs_theorem}{}\subsection*{{Mnev's theorem}}\label{mnevs_theorem}

Mn\"e{}v's universality theorem says that any [[semialgebraic set]] in $\mathbb{R}^n$ defined over integers is stably equivalent to the realization space of some oriented matroid.

\hypertarget{categories_of_matroids}{}\subsection*{{Categories of matroids}}\label{categories_of_matroids}

To be written, possibly with some original research.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Matroid}{matroid}, \href{http://en.wikipedia.org/wiki/Mnev%27s_universality_theorem}{Mn\"e{}v's universality theorem}, \href{http://en.wikipedia.org/wiki/Oriented_matroid}{oriented matroid}
\item Hassler Whitney, \emph{On the abstract properties of linear dependence}, American Journal of Mathematics (The Johns Hopkins University Press) 57 (3): 509--533, 1935, \href{http://jstor.org/stable/2371182}{jstor}, \href{http://www.ams.org/mathscinet-getitem?mr=1507091}{MR1507091}
\item J. Oxley, \emph{What is a matroid}, \href{http://www.math.lsu.edu/~oxley/survey4.pdf}{pdf}
\item James G. Oxley, \emph{Matroid theory}, Oxford Grad. Texts in Math. 1992, 2010
\item some papers on Coxeter matroids \href{http://www.math.ufl.edu/~white/cox.html}{html}
\item MathOverflow question \href{http://mathoverflow.net/questions/72154/mnevs-universality-corollaries-quantitative-versions}{Mn\"e{}v's universality corollaries, quantitative versions}
\item Eric Katz, Sam Payne, \emph{Realization space for [[tropical geometry|tropical]] fans}, \href{http://users.math.yale.edu/~sp547/pdf/Realization-spaces.pdf}{pdf}
\item Nikolai E. Mnev, \emph{The universality theorems on the classification problem of configuration varieties and convex polytopes varieties}, pp. 527-543, in ``Topology and geometry: Rohlin Seminar.'' Edited by O. Ya. Viro. Lecture Notes in Mathematics, \textbf{1346}, Springer 1988; \emph{A lecture on universality theorem} (in Russian) \href{http://club.pdmi.ras.ru/~panina/9.pdf}{pdf}
\item Talal Ali Al-Hawary, \emph{Free objects in the category of geometries}, \href{http://www.emis.de/journals/HOA/IJMMS/Volume26_12/770.pdf}{pdf}
\item Talal Ali Al-Hawary, D. George McRae, \emph{Toward an elementary axiomatic theory of the category of LP-matroids}, Applied Categorical Structures \textbf{11}: 157--169, 2003, \href{http://dx.doi.org/10.1023/A:1023557229668}{doi}
\item Hirokazu Nishimura, Susumu Kuroda, \emph{A lost mathematician, Takeo Nakasawa: the forgotten father of matroid theory} 1996, 2009
\item William H. Cunningham, \emph{Matching, matroids, and extensions}, Math. Program., Ser. B \textbf{91}: 515--542 (2002) \href{http://dx.doi.org/10.1007/s101070100256}{doi}
\item L. Lov\'a{}sz, \emph{Matroid matching and some applications}, J. Combinatorial Theory B \textbf{28}, 208--236 (1980)
\item A. Bj\"o{}rner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, \emph{Oriented matroids}, Cambridge Univ. Press 1993, 2000, \href{http://reslib.com/book/Oriented_Matroids}{view at reslib.com}
\item Matthew Baker, \emph{Matroids over hyperfields}, \href{http://arxiv.org/abs/1601.01204}{arxiv/1601.01204}
\item David Marker, \emph{Model Theory: An Introduction}, Graduate Texts in Math. 217, Springer-Verlag New York, 2002.
\item Tom Braden, Carl Mautner, \emph{Matroidal Schur algebras}, \href{http://arxiv.org/abs/1609.04507}{arxiv/1609.04507}
\item P. S. Kung, \emph{A Source Book in Matroid Theory}, Birkh\"a{}user, Boston, 1986.

\end{itemize}
[[!redirects matroids]] [[!redirects pregeometry]] [[!redirects pregeometries]]



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