Definition:Matroid

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Definition

Let $M = \struct{S,\mathscr I}$ be an independence system.


Definition 1

$M$ is called a matroid on $S$ if $M$ also satisfies:

\((\text I 3)\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I \)             


Definition 2

$M$ is called a matroid on $S$ if $M$ also satisfies:

\((\text I 3')\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I \)             


Definition 3

$M$ is called a matroid on $S$ if $M$ also satisfies:

\((\text I 3'')\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{ \size {V \cup Z} = \size U} \)             


Definition 4

$M$ is called a matroid on $S$ if $M$ also satisfies:

\((\text I 3''')\)   $:$     \(\displaystyle \forall A \subseteq S:\) \(\displaystyle \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} \)             


When the context is obvious, $M = \struct{S, \mathscr I}$ is simply called a matroid.

Independent Set

An element of $\mathscr I$ is called an independent set of $M$.


Dependent Set

A subset of $S$ that is not an element of $\mathscr I$ is called a dependent set of $M$.


Examples of Matroids

Uniform Matroid

Let $S$ be a finite set of cardinality $n$.

Let $\mathscr I_{k,n}$ be the set of all subsets of $S$ of cardinality less than or equal to $k$.


Then the ordered pair $\struct {S, \mathscr I_{k, n} }$ is called the uniform matroid of rank $k$ and is denoted $U_{k,n}$.


Free Matroid

Let $S$ be a finite set.

Let $\mathscr I = \powerset S$ be the power set of $S$.

That is, let $\mathscr I$ be the set of all subsets of $S$:

$\mathscr I := \set {X: X \subseteq S}$


Then the ordered pair $\struct{S, \mathscr I}$ is called the free matroid of $S$.


Matroid Induced by Linear Independence in Vector Space

Let $V$ be a vector space.

Let $S$ be a finite subset of $V$.

Let $\mathscr I$ be the set of linearly independent subsets of $S$.


Then the ordered pair $\struct{S, \mathscr I}$ is called a matroid induced on $S$ by linear independence in $V$.


Cycle Matroid

Let $G$ be a graph.

Let $E$ be the edge set of $G$.

Let $\mathscr I$ be the set of edge sets of subgraphs of $G$ that contain no cycles.


Then the ordered pair $\struct{E, \mathscr I}$ is called the cycle matroid of the graph $G$.


Matroid Induced by Algebraic Independence

Let $L / K$ be a field extension.

Let $S \subseteq L$ be a finite subset of $L$.

Let $\mathscr I$ be the set of algebraically independent subsets of $S$.


Then $\struct {S, \mathscr I}$ is called the matroid induced by algebraic independence over $K$ on $S$.


Matroid Induced by Affine Independence

Let $\R^n$ be the $n$-dimensional real Euclidean space.

Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$.

Let $\mathscr I$ be the set of affinely independent subsets of $S$.


Then $\struct{S, \mathscr I}$ is called the matroid induced by affine independence on $S$.


Matroid Induced by Linear Independence in Abelian Group

Let $\struct{G, +}$ be a torsion-free Abelian group.

Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$.

Let $S$ be a finite subset of $G$.

Let $\mathscr I$ be the set of linearly independent subsets of $S$.


Then the ordered pair $\struct{S, \mathscr I}$ is called the matroid induced by linear independence in $G$ on $S$.