In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent...
11 KB (1,636 words) - 07:03, 28 December 2023
In combinatorics, a branch of mathematics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector...
60 KB (8,752 words) - 21:40, 19 August 2024
has a basis, then the axiom of choice is true. Thus the two assertions are equivalent. Basis of a matroid Basis of a linear program Change of basis – Coordinate...
34 KB (4,740 words) - 19:32, 4 September 2024
Basis of a matroid Generating set of an ideal: Gröbner basis Hilbert's basis theorem Generating set of a group Base (topology) Change of basis Greedoid...
2 KB (296 words) - 12:20, 25 October 2020
of matroid they are using. For instance, given an independence oracle for any matroid, it is possible to find the minimum weight basis of the matroid...
33 KB (4,332 words) - 13:26, 29 November 2023
theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite...
16 KB (2,269 words) - 12:47, 5 July 2024
a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is...
8 KB (1,409 words) - 01:59, 19 June 2022
mathematics, a delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid...
4 KB (514 words) - 20:40, 6 July 2021
In matroid theory, the dual of a matroid M {\displaystyle M} is another matroid M ∗ {\displaystyle M^{\ast }} that has the same elements as M {\displaystyle...
8 KB (950 words) - 13:11, 1 February 2023
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane...
31 KB (3,970 words) - 08:05, 17 June 2024
Geometric lattice (redirect from Matroid lattice)
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular...
8 KB (1,190 words) - 18:39, 31 January 2024
mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid. For any two bases A {\displaystyle...
8 KB (1,189 words) - 10:29, 11 May 2023
a uniform matroid. The unique basis of this matroid is the ground-set itself, E. Among matroids on E, the free matroid on E has the most independent sets...
2 KB (259 words) - 06:55, 1 February 2024
Matroid partitioning is a problem arising in the mathematical study of matroids and in the design and analysis of algorithms. Its goal is to partition...
13 KB (2,048 words) - 20:31, 20 March 2023
In combinatorics, a branch of mathematics, a weighted matroid is a matroid endowed with a function that assigns a weight to each element. Formally, let...
9 KB (1,652 words) - 00:44, 7 May 2024
mathematical theory of matroids, a paving matroid is a matroid in which every circuit has size at least as large as the matroid's rank. In a matroid of rank r {\displaystyle...
7 KB (911 words) - 10:42, 3 May 2023
In combinatorics, a matroid embedding is a set system (F, E), where F is a collection of feasible sets, that satisfies the following properties. Accessibility...
2 KB (195 words) - 19:28, 31 October 2022
In matroid theory, a Sylvester matroid is a matroid in which every pair of elements belongs to a three-element circuit (a triangle) of the matroid. The...
3 KB (410 words) - 20:04, 18 June 2022
In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer...
6 KB (944 words) - 22:21, 18 July 2020
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota...
8 KB (983 words) - 06:31, 17 December 2023
a colored matroid is a matroid whose elements are labeled from a set of colors, which can be any set that suits the purpose, for instance the set of the...
2 KB (195 words) - 20:09, 19 December 2023
Greedoid (category Families of sets)
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study...
12 KB (1,729 words) - 07:31, 20 June 2024
theory of matroids, a minor of a matroid M is another matroid N that is obtained from M by a sequence of restriction and contraction operations. Matroid minors...
16 KB (1,989 words) - 01:01, 28 May 2024
matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose...
7 KB (819 words) - 18:30, 17 June 2022
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which...
7 KB (965 words) - 14:27, 8 November 2021
K-set (geometry) (category Matroid theory)
minimum weight basis of the matroid for each possible value of λ {\displaystyle \lambda } . If one graphs the weight functions as lines in a plane, the k...
16 KB (1,877 words) - 00:12, 3 January 2024
theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is,...
10 KB (1,429 words) - 13:28, 21 September 2023
Dimension (vector space) (redirect from Linear algebra/Dimension of a vector space)
V).} A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length of a module...
9 KB (1,485 words) - 00:27, 14 May 2024
combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid. The problem was formulated...
20 KB (2,862 words) - 10:21, 6 January 2024
Transcendental extension (redirect from Transcendence basis)
examples of finitary matroids (pregeometries). Any finitary matroid has a basis, and all bases have the same cardinality. If G is a generating set of L (i...
12 KB (1,679 words) - 09:36, 7 August 2024