A matroid is an abstract mathematical object which encodes the notion of independence. It has relevant applications in graph theory, linear algebra, geometry, topology, network theory, and more. Matroid theory is a thriving area of research.

The simplest way to construct a matroid is via a matrix. Matroids constructed this way are called linear or representable.

A := Matrix([[1,-1,0,1],[1,1,1,0],[1,1,0,1]]);

$A≔\left[\begin{array}{cccc}1& \mathrm{-1}& 0& 1\\ 1& 1& 1& 0\\ 1& 1& 0& 1\end{array}\right]$

with(Matroids);

$\left[\mathrm{AreIsomorphic}\,\mathrm{Bases}\,\mathrm{CharacteristicPolynomial}\,\mathrm{Circuits}\,\mathrm{Contraction}\,\mathrm{Deletion}\,\mathrm{DependentSets}\,\mathrm{Dual}\,\mathrm{ExampleMatroids}\,\mathrm{Flats}\,\mathrm{GroundSet}\,\mathrm{Hyperplanes}\,\mathrm{IndependentSets}\,\mathrm{IsMinorOf}\,\mathrm{Matroid}\,\mathrm{Rank}\,\mathrm{SetDisplayStyle}\,\mathrm{TuttePolynomial}\right]$

M := Matroid(A);

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{cccc}1& \mathrm{-1}& 0& 1\\ 1& 1& 1& 0\\ 1& 1& 0& 1\end{array}\right]\end{array}⟩$

This matroid encodes the linear dependencies among the columns of $A$. The so-called ground set of the matroid consists of the numbers 1 through 4, interpreted as column indices into $A$.

We can ask for which subsets of columns are:

linearly independent,

linearly dependent, and

bases for the column space of A.

IndependentSets(M);

$\left[\varnothing \,\left\{1\right\}\,\left\{2\right\}\,\left\{3\right\}\,\left\{4\right\}\,\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\,\left\{1\,2\,3\right\}\,\left\{1\,2\,4\right\}\,\left\{2\,3\,4\right\}\right]$

DependentSets(M);

$\left[\left\{1\,3\,4\right\}\,\left\{1\,2\,3\,4\right\}\right]$

Bases(M);

$\left[\left\{1\,2\,3\right\}\,\left\{1\,2\,4\right\}\,\left\{2\,3\,4\right\}\right]$

These answers change if the column vectors are considered over a finite field, e.g. the field with two elements:

Mmodular := Matroid(A,2);

$\mathrm{Mmodular}≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{cccc}1& 1& 0& 1\\ 1& 1& 1& 0\\ 1& 1& 0& 1\end{array}\right]\mathbf{mod}2\end{array}⟩$

Bases(Mmodular);

$\left[\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\right]$

Notice that the size of a basis changed from 3 to 2. This number is the rank of the matroid, which agrees with the familiar notion of rank (of the column space).

Rank(M);

$3$

Rank(Mmodular);

$2$

Matroids are much more general than this! As an abstraction of independence, matroids also encode graph independence.

Given a graph G, a subset of its edges are called dependent if they contain a path which forms a closed loop, known as a circuit.

with(GraphTheory):

G := Graph({{a,b},{a,c},{b,d},{a,d}});

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 4\; vertices\; and\; 4\; edge(s)}$

GraphicMatroid := Matroid(G);

$\mathrm{GraphicMatroid}≔⟨\begin{array}{c}\mathrm{th\ⅇ\; graphic\; matroi\ⅆ\; on\; th\ⅇ\; graph:}\\ \mathrm{PLOT}\left(\mathrm{...}\right)\end{array}⟩$

Circuits(GraphicMatroid);

$\left[\left\{''a\_b''\,''a\_d''\,''b\_d''\right\}\right]$

Inspired by linear algebra, one may take the definition of a basis as a maximal independent set. The bases of a graphic matroid are its spanning forests.

Bases(GraphicMatroid);

$\left[\left\{''a\_b''\,''a\_c''\,''a\_d''\right\}\,\left\{''a\_b''\,''a\_c''\,''b\_d''\right\}\,\left\{''a\_c''\,''a\_d''\,''b\_d''\right\}\right]$

In fact, every concept about linear independence coming from linear algebra (rank, bases, etc) can be axiomatized and interpreted for a graphic matroid.

Conversely, the concept of a circuit from graph theory applies to linear matroids.

Rank(GraphicMatroid);

$3$

Circuits(M);

$\left[\left\{1\,3\,4\right\}\right]$

Circuits(Mmodular);

$\left[\left\{1\,2\right\}\,\left\{1\,3\,4\right\}\,\left\{2\,3\,4\right\}\right]$

This is the power of the abstraction of matroids. One rigorous definition of a matroid is as follows.

A matroid is a pair $M\=\left(E\,I\right)$, where

$E$ is a finite set called the ground set and

$\mathrm{I}$ is a collection of subsets of $E$ called independent sets which satisfy the axioms:

(Axiom 1) The empty set is an independent set.

(Axiom 2) Every subset of an independent set is independent.

(Axiom 3) If $\mathrm{I1}$ and $\mathrm{I2}$ are independent sets and $\mathrm{I1}$ has more elements than $\mathrm{I2}$, then there exists an element of $\mathrm{I2}$ which when included in $\mathrm{I1}$ results in an independent set.

The matroid package includes functionality for constructing a matroid directly from its independent sets:

AxiomaticMatroid := Matroid([1,2,3], independentsets = [{},{1},{2},{3},{1,3},{2,3}]);

$\mathrm{AxiomaticMatroid}≔⟨\mathrm{a\; matroi\ⅆ\; on\; 3\; \ⅇl\ⅇm\ⅇnts\; with\; 5\; in\ⅆ\ⅇp\ⅇn\ⅆ\ⅇnt\; s\ⅇts}⟩$

In fact, for each of the matroid properties of independent sets, bases, dependent sets, and circuits we have seen, one may construct a matroid (provided they satisfy certain axioms, listed on the Matroid help page).

Each property uniquely determines the rest, and the matroids package supports several other axiomatic constructions (via flats, hyperplanes, or a rank function).

Algorithms which convert between these representations are called cryptomorphisms. The matroids package showcases fast implementations of these algorithms.

Circuits(AxiomaticMatroid);

$\left[\left\{1\,2\right\}\right]$

Bases(AxiomaticMatroid);

$\left[\left\{1\,3\right\}\,\left\{2\,3\right\}\right]$

Beyond linear matroids constructed from a matrix, graphic matroids constructed from a graph, and general matroids constructed via axioms, the matroid package also features the construction of algebraic matroids, created from polynomial ideals.

with(PolynomialIdeals):

AlgebraicMatroid := Matroid(<x+y+z^2,z^2+y>);

$\mathrm{AlgebraicMatroid}≔⟨\begin{array}{c}\mathrm{th\&ExponentialE;\; alg\&ExponentialE;braic\; matroi\&DifferentialD;\; on\; th\&ExponentialE;\; polynomial\; i\&DifferentialD;\&ExponentialE;al:}\\ ⟨z^2+y\&comma;z^2+x+y⟩\end{array}⟩$

DependentSets(AlgebraicMatroid);

$\left[\left\{1\right\}\&comma;\left\{1\&comma;2\right\}\&comma;\left\{1\&comma;3\right\}\&comma;\left\{2\&comma;3\right\}\&comma;\left\{1\&comma;2\&comma;3\right\}\right]$

That $\left\{1\right\}$ is a dependent set indicates that there exists a polynomial in the ideal which involves only the first variable, $x$.

The matroids package features a gallery of well-known matroids, which can be made available by loading the ExampleMatroids subpackage.

with(ExampleMatroids);

$\left[\mathrm{Fano}\&comma;\mathrm{Hesse}\&comma;\mathrm{MacLane}\&comma;\mathrm{NCubeMatroid}\&comma;\mathrm{NonFano}\&comma;\mathrm{NonPappus}\&comma;\mathrm{Pappus}\&comma;\mathrm{TicTacToe}\&comma;\mathrm{UniformMatroid}\&comma;\mathrm{Vamos}\right]$

Additionally, one may perform several operations on matroids:

AreIsomorphic: determine if two matroids are the same, under some relabeling of the ground set;

Deletion and Contraction: generalizations of deletion and contraction of edges of a graph;

Dual: a generalization of the dual of a planar graph. Unlike for graphs, duals of matroids always exist. For linear matroids, duality corresponds to orthogonal complements of the row space.

TuttePolynomial and CharacteristicPolynomial: polynomial invariants of matroids which generalize those of a graph;

IsMinorOf: a test to check if one matroid can be obtained by another via a sequence of deletions and contractions.

ContractionMatroid := Contraction(GraphicMatroid,{4});

$\mathrm{ContractionMatroid}≔⟨\mathrm{a\; matroi\&DifferentialD;\; on\; 4\; \&ExponentialE;l\&ExponentialE;m\&ExponentialE;nts\; with\; 1\; circuit}⟩$

AreIsomorphic(ContractionMatroid,AxiomaticMatroid);

$\mathrm{false}$

IsMinorOf(ContractionMatroid,GraphicMatroid);

$\mathrm{true},\varnothing ,\varnothing$

Dual(M);

$⟨\mathrm{a\; matroi\&DifferentialD;\; on\; 4\; \&ExponentialE;l\&ExponentialE;m\&ExponentialE;nts\; with\; 3\; bas\&ExponentialE;s\; of\; siz\&ExponentialE;\; 1}⟩$

Matroids:-TuttePolynomial(GraphicMatroid,x,y);

$x^3+x^2+xy$

Matroids:-CharacteristicPolynomial(GraphicMatroid,k);

$k^3-4k^2+5k-2$

The Hypergraphs package is the computational backbone of the matroids package, and it is much more than that!

A hypergraph is a pair $\left(V\&comma;E\right)$ consisting of a finite set $V$ called vertices and a collection $E$ of subsets of $V$ called hyperedges.

Hypergraphs, as indicated by the name, generalize graphs: a graph can be thought of as a hypergraph where every hyperedge has size two (or size one if self-loops are allowed).

We create a hypergraph with the Hypergraph command.

with(Hypergraphs);

$\left[\mathrm{AddHyperedges}\&comma;\mathrm{AddVertices}\&comma;\mathrm{AntiRank}\&comma;\mathrm{AreEqual}\&comma;\mathrm{AreIsomorphic}\&comma;\mathrm{ComplementHypergraph}\&comma;\mathrm{DegreeProfile}\&comma;\mathrm{Draw}\&comma;\mathrm{DualHypergraph}\&comma;\mathrm{ExampleHypergraphs}\&comma;\mathrm{Hyperedges}\&comma;\mathrm{Hypergraph}\&comma;\mathrm{IsConnected}\&comma;\mathrm{IsEdge}\&comma;\mathrm{IsLinear}\&comma;\mathrm{IsRegular}\&comma;\mathrm{IsUniform}\&comma;\mathrm{LineGraph}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{NumberOfHyperedges}\&comma;\mathrm{NumberOfVertices}\&comma;\mathrm{PartialHypergraph}\&comma;\mathrm{Rank}\&comma;\mathrm{SubHypergraph}\&comma;\mathrm{Transversal}\&comma;\mathrm{VertexEdgeIncidenceGraph}\&comma;\mathrm{Vertices}\right]$

H := Hypergraph([1,2,3,4],[{1,2},{1,3},{2,3,4}]);

$H≔\mathrm{<\; a\; hypergraph\; on\; 4\; vertices\; with\; 3\; hyperedges\; >}$

For few vertices and hyperedges, one can visualize a hypergraph as an augmented graph.

Distinguished nodes of the graph correspond to vertices of the hypergraph. Pairs of nodes are connected, as usual, if they form a (hyper)edge.

Additional, auxiliary nodes are included for every hyperedge of size greater than two and auxiliary edges connect such nodes with the vertices they include.

Draw(H);

Procedures for manipulating hypergraphs include AddHyperedges and AddVertices.

Given a hypergraph, the functions ComplementHypergraph, DualHypergraph, and SubHypergraph create new hypergraphs in the ways the names suggest.

Basic functionality such as Hyperedges, NumberOfHyperedges, Vertices, and NumberOfVertices are available, as are simple queries including AreEqual, IsConnected, and IsEdge.

The functions DegreeProfile and VertexEdgeIncidenceGraph directly generalize those notions from graphs to hypergraphs.

H2 := AddHyperedges(AddVertices(H,["apple"]),[{1,4},{2,"apple",3,4},{3}]);

$\mathrm{H2}≔\mathrm{<\; a\; hypergraph\; on\; 5\; vertices\; with\; 6\; hyperedges\; >}$

Draw(H2);

[AreEqual(H,H2), IsEdge(H2,{2,1}), NumberOfHyperedges(H2), Hypergraphs:-NumberOfVertices(H2), Hypergraphs:-IsConnected(H2), DegreeProfile(H)];

$\left[\mathrm{false}\&comma;\mathrm{true}\&comma;6\&comma;5\&comma;\mathrm{true}\&comma;\left[2\&comma;2\&comma;2\&comma;1\right]\right]$

The major advancement in Maple with the hypergraphs package has to do with what goes on behind the scenes.

Subsets are carefully encoded using bit-vectors to make hefty calculations fast and feasible.

with(ExampleHypergraphs);

$\left[\mathrm{Fan}\&comma;\mathrm{Kuratowski}\&comma;\mathrm{Lovasz}\&comma;\mathrm{NonEmptyPowerSet}\&comma;\mathrm{RandomHypergraph}\right]$

Below, we illustrate the core hypergraph algorithms on a random hypergraph on 10 vertices with 100 hyperedges.

R := RandomHypergraph(10,100);

$R≔\mathrm{<\; a\; hypergraph\; on\; 10\; vertices\; with\; 100\; hyperedges\; >}$

Draw(R);

The Min function computes the hyperedges which do not properly contain another hyperedge.

The Max function computes those which are not properly contained in another hyperedge.

The Transversal function computes the sets of vertices for which every hyperedge contains some element in that set.

Hyperedges(Min(R));

$\left[\left\{6\&comma;7\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;10\right\}\&comma;\left\{7\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;5\right\}\&comma;\left\{1\&comma;4\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;4\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;7\&comma;8\right\}\&comma;\left\{2\&comma;3\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;9\right\}\&comma;\left\{2\&comma;4\&comma;5\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;8\&comma;9\right\}\&comma;\left\{3\&comma;5\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;10\right\}\&comma;\left\{1\&comma;4\&comma;5\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;5\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;10\right\}\&comma;\left\{2\&comma;5\&comma;6\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;7\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;8\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;8\&comma;10\right\}\&comma;\left\{4\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{6\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;4\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;6\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;6\&comma;7\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;5\&comma;8\right\}\&comma;\left\{2\&comma;4\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;6\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;7\&comma;9\right\}\&comma;\left\{3\&comma;4\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;9\&comma;10\right\}\right]$

Hyperedges(Max(R));

$\left[\left\{2\&comma;4\&comma;5\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;5\&comma;6\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;5\&comma;7\right\}\&comma;\left\{2\&comma;4\&comma;5\&comma;7\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;7\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;6\&comma;7\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;5\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;5\&comma;6\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;6\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;6\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;4\&comma;5\&comma;6\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;6\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;6\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;7\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;6\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;4\&comma;6\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;5\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;6\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;5\&comma;7\&comma;9\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;6\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;4\&comma;5\&comma;7\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;6\&comma;7\&comma;8\&comma;9\&comma;10\right\}\right]$

Hyperedges(Transversal(R));

$\left[\left\{3\&comma;4\&comma;6\&comma;10\right\}\&comma;\left\{3\&comma;5\&comma;6\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;7\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;7\&comma;10\right\}\&comma;\left\{3\&comma;5\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;7\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;4\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;5\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;5\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;6\&comma;7\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;6\&comma;7\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;3\&comma;5\&comma;7\&comma;8\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{2\&comma;4\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{3\&comma;4\&comma;6\&comma;7\&comma;8\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;6\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;6\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;6\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;7\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;8\&comma;9\right\}\&comma;\left\{2\&comma;3\&comma;4\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;8\&comma;9\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;8\&comma;9\right\}\&comma;\left\{3\&comma;4\&comma;6\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;6\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;7\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;5\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{4\&comma;5\&comma;6\&comma;8\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{4\&comma;6\&comma;7\&comma;8\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;3\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;4\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;5\&comma;9\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;5\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;2\&comma;6\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;6\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;4\&comma;7\&comma;9\&comma;10\right\}\&comma;\left\{4\&comma;5\&comma;7\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;3\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{3\&comma;4\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;5\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{4\&comma;5\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{1\&comma;6\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{5\&comma;6\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;7\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{4\&comma;7\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{5\&comma;7\&comma;8\&comma;9\&comma;10\right\}\&comma;\left\{2\&comma;3\&comma;5\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{2\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\right\}\&comma;\left\{1\&comma;2\&comma;4\&comma;5\&comma;6\&comma;10\right\}\right]$

Put another way, consider the hypergraph $\mathrm{Food}$ whose vertices are ingredients in your kitchen, and whose hyperedges are recipes.

Then $\mathrm{Min}\left(\mathrm{Food}\right)$ are those recipes which require a minimal set of ingredients (i.e. removing any ingredient prevents any recipe from being made).

$\mathrm{Max}\left(\mathrm{Food}\right)$ are those recipes which maximally use ingredients (i.e. you cannot include an additional ingredient to make a bigger recipe).

$\mathrm{Transversal}\left(\mathrm{Food}\right)$ are all sets of ingredients an adversary could steal from your fridge which would prevent you from making any recipe.

In the context of matroids, the sets of subsets that can be used to define a matroid axiomatically are all hypergraphs, and they are stored as such if they are known for a given matroid. Several cryptomorphisms come directly from these hypergraph operations. For example, the Circuits of a matroid $M$ are just $\mathrm{Min}\left(\mathrm{DependentSets}\left(M\right)\right)$.

Below, we illustrate the remaining functionality and invite you to check out the details on our help pages!

DrawGraph(Hypergraphs:-LineGraph(H));

[Rank(H),AntiRank(H)];

$\left[3\&comma;2\right]$

[IsLinear(H),IsRegular(H),IsUniform(H)];

$\left[\mathrm{true}\&comma;\mathrm{false}\&comma;\mathrm{false}\right]$

with(ExampleHypergraphs);

$\left[\mathrm{Fan}\&comma;\mathrm{Kuratowski}\&comma;\mathrm{Lovasz}\&comma;\mathrm{NonEmptyPowerSet}\&comma;\mathrm{RandomHypergraph}\right]$

[Draw(Kuratowski({1,2,3,4,5},2)),Draw(Kuratowski({1,2,3,4},3))];

$\left[\mathrm{PLOT}\left(\mathrm{...}\right)\&comma;\mathrm{PLOT}\left(\mathrm{...}\right)\right]$

Draw(Lovasz(5));

NumberOfHyperedges(Lovasz(5));

$206$