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Abstraction of ordered linear algebra
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane
Oriented_matroid
Pseudolines arranged largely to study arrangements of lines
connected flip graph. Each rank-3 oriented matroid is equivalent to an arrangement of pseudolines, and each oriented matroid which is also uniform (in which
Arrangement_of_pseudolines
Realization of semialgebraic sets by points
represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point
Mnëv's_universality_theorem
Abstraction of linear independence of vectors
In combinatorics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many
Matroid
Minkowsi sum of line segments
cocircuits of M {\displaystyle {\mathcal {M}}} and if we consider the oriented matroid M {\displaystyle {\mathcal {M}}} represented by W {\displaystyle \mathbf
Zonotope
Method for linear optimization
termed "Bland oriented matroids" by Jack Edmonds. Another pivoting rule, the criss-cross algorithm, avoids cycles on all oriented-matroid linear-programs
Bland's_rule
Branch of geometry that studies combinatorial properties and constructive methods
Configurations Line arrangements Hyperplane arrangements Buildings An oriented matroid is a mathematical structure that abstracts the properties of directed
Discrete_geometry
Partition of space by hyperplanes
discussion, but it makes no material difference. Supersolvable arrangement Oriented matroid "Arrangement of hyperplanes", Encyclopedia of Mathematics, EMS Press
Arrangement_of_hyperplanes
Matroid with no linear representation
another geometric lattice of the same rank. The Vámos matroid can be oriented. In oriented matroids, a form of the Hahn–Banach theorem follows from a certain
Vámos_matroid
Graph with sign-labeled edges
are two matroids associated with a signed graph, called the signed-graphic matroid (also called the frame matroid or sometimes bias matroid) and the
Signed_graph
Method for mathematical optimization
based on his previous papers on oriented-matroid theory. However, Bland's rule exhibits cycling on some oriented-matroid linear-programming problems. The
Criss-cross_algorithm
Existence of a line through two points
of a rank-3 oriented matroid. The points and lines of geometries defined using other number systems than the real numbers also form matroids, but not necessarily
Sylvester–Gallai_theorem
Concept in mathematical optimisation
In mathematical optimization, Cunningham's rule (also known as least recently considered rule or round-robin rule) is an algorithmic refinement of the
Cunningham's_rule
Graph representing faces of another graph
2010.01.018, MR 2601261. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10
Dual_graph
Matroid with graph forests as independent sets
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the
Graphic_matroid
Refinement of the simplex method for linear optimization
In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization
Zadeh's_rule
American mathematician and politician (born 1977)
(2007). "On D.K. Biss' papers "The homotopy type of the matroid Grassmannian" and "Oriented matroids, complex manifolds, and a combinatorial model for BU""
Daniel_Biss
Convex polyhedron projected from hypercube
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon)
Zonohedron
On when a space equals the closed convex hull of its extreme points
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector
Krein–Milman_theorem
Japanese mathematician (born 1951)
for his contributions to optimization, polyhedral computation and oriented matroid theory. Fukuda is a professor in optimization and computational geometry
Komei_Fukuda
Quadratic programming as a special case
Such LCPs can be solved when they are formulated abstractly using oriented-matroid theory. Complementarity theory Physics engine Impulse/constraint type
Linear complementarity problem
Linear_complementarity_problem
Ternary relation on points in the plane
correspondence between CC systems and uniform acyclic oriented matroids of rank 3. These matroids in turn have a 1-1 correspondence to topological equivalence
CC_system
Abstract geometry without 2-point lines
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. In
Sylvester_matroid
Geometric structure of 8 points and 8 lines
points. As a matroid, it has been called the MacLane matroid, after the work of Saunders MacLane (1936) proving that it cannot be oriented; it is one of
Möbius–Kantor_configuration
Award for advancements in discrete mathematics
semialgebraic set is equivalent to the space of realizations of an oriented matroid. 1994: Louis Billera for finding bases of piecewise-polynomial function
Fulkerson_Prize
Operation combining two oriented knots
pp. 125–136. Ramirez Alfonsin, J. L. (1999), "Spatial graphs and oriented matroids: the trefoil", Discrete and Computational Geometry, 22 (1): 149–158
Knot_(mathematics)
Largest independent set of paired elements
combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid, a structure that abstracts
Matroid_parity_problem
Subroutine for testing independence
mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure
Matroid_oracle
Directed graph where edges have a capacity
(computer networking) Flow graph (disambiguation) Max-flow min-cut theorem Oriented matroid Shortest path problem Nowhere-zero flow Active flow network A.V. Goldberg
Flow_network
Method to solve optimization problems
algorithm – Method of computing optimal strategies for last-success problems Oriented matroid – Abstraction of ordered linear algebra Quadratic programming – Solving
Linear_programming
Mathematical structure
simple directed graphs by allowing multiple same-oriented edges between pairs of vertices. Matroids are a quite general mathematical abstraction that
Sparsity_matroid
Algorithm for linear programming
Murty (1983, p. 79) There are abstract optimization problems, called oriented matroid programs, on which Bland's rule cycles (incorrectly) while the criss-cross
Simplex_algorithm
Algebraic encoding of graph connectivity
1016/j.tcs.2004.02.023. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10
Tutte_polynomial
Convex hull of a finite set of points in a Euclidean space
have a bit-length which is not polynomial in this representation. Oriented matroid Nef polyhedron Steinitz's theorem for convex polyhedra Branko Grünbaum
Convex_polytope
American mathematician
Monotone operator (Cyclic decomposition of maximal monotone operator) Oriented matroids (realizable OMs and applications) Carathéodory's theorem (convex hull)
R._Tyrrell_Rockafellar
German American mathematician
Berkeley Max Planck Institute for Mathematics in the Sciences Thesis Oriented Matroids and Combinatorial Convex Geometry; Computational Synthetic Geometry
Bernd_Sturmfels
Pencil and paper connection game
Versions of the Shannon switching game played on a directed graph and an oriented matroid have been described for theoretical purposes; but no corresponding
Shannon_switching_game
American mathematician
of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids"
Jon_Folkman
Flat-sided three-dimensional shape
Bokowski, J.; Guedes de Oliveira, A. (2000), "On the generation of oriented matroids", Discrete and Computational Geometry, 24 (2–3): 197–208, doi:10.1007/s004540010027
Polyhedron
Embedding of a Grassmannian into projective space
Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications, vol. 46 (2nd ed
Plücker_embedding
Partition of graph into sequence of paths
efficient graph algorithms. They may also be generalized from graphs to matroids. Several important classes of graphs may be characterized as the graphs
Ear_decomposition
American mathematician
"It seems fair to say that the major credit for the origination of oriented matroid theory should be shared by Robert Bland, Jon Folkman, Michel Las Vergnas
Robert_G._Bland
Toroidal polyhedron with 14 triangle faces
Bokowski, J.; Guedes de Oliveira, A. (2000), "On the Generation of Oriented Matroids", Discrete & Computational Geometry, 24: 197–208, doi:10.1007/s004540010027
Császár_polyhedron
1112/blms/18.6.571, MR 0859948 Ramírez Alfonsín, J. L. (2001), "Lawrence oriented matroids and a problem of McMullen on projective equivalences of polytopes"
McMullen_problem
Element of graph theory
ISBN 978-0-521-59840-8, MR 1477750. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10
Acyclic_orientation
(2001-07-01). "Cocircuit Graphs and Efficient Orientation Reconstruction in Oriented Matroids". Eur. J. Comb. 22 (5): 587–600. doi:10.1006/eujc.2001.0481. ISSN 0195-6698
Graph_of_a_polytope
Swedish mathematician (born 1947)
Michael; Sturmfels, Bernd; White, Neil; Ziegler, Günter M. (1999). Oriented Matroids (2nd ed.). Cambridge University Press. ISBN 0-521-77750-X. Björner
Anders_Björner
French mathematician
Beginning in 1975, he became one of the pioneers of the theory of oriented matroids, and since that time he was interested in connections between combinatorics
Michel_Las_Vergnas
Relation on disjoint pairs of sets
framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any countable category is an induced subcategory of
Separoid
(2007). "On D.K. Biss' papers 'The homotopy type of the matroid Grassmannian' and 'Oriented matroids, complex manifolds, and a combinatorial model for BU'"
List_of_incomplete_proofs
_{2}} -graded sets. Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the
Signed_set
Smallest convex set containing a given set
spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids. It is not obvious that the first definition makes sense: why should
Convex_hull
Graph with at most one cycle per component
minimum k such that the edges of G can be oriented to form a directed graph with outdegree at most k. Due to the matroid structure of pseudoforests, the pseudoarboricity
Pseudoforest
affine Gale diagrams can also be described through the duality of oriented matroids. As with the linear diagram, a subset of vertices forms a face if
Gale_diagram
and Gutman and Borovićanin (2011). In the matroid theory the nullity of the graph is the nullity of the oriented incidence matrix M associated with the graph
Nullity_(graph_theory)
Oriented projective geometry is an oriented version of real projective geometry. Whereas the real projective plane describes the set of all unoriented
Oriented_projective_geometry
Hungarian mathematician (born 1955)
independently published on the criss-cross algorithm. The theory of oriented matroids has also been used by Terlaky and Zhang (1991) to prove that their
Tamás_Terlaky
American/Canadian mathematician and computer scientist
he proved the matroid intersection theorem, a very general combinatorial min-max theorem which, in modern terms, showed that the matroid intersection problem
Jack_Edmonds
The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids. A 2025 article by Zelin Dong, Fenglei Fan, Huan Xiong, and Tieyong
Vertex_enumeration_problem
American geometer (1933–2021)
in the study of arrangements of pseudolines and (more generally) oriented matroids. His work with Pollack includes such results as the first nontrivial
Jacob_E._Goodman
On the number of spanning trees in a graph
regular matroids, a generalization of the graphic matroids (Maurer 1976). Kirchhoff's theorem can be modified to give the number of oriented spanning
Kirchhoff's_theorem
Characteristic of undirected graphs
the rank of the oriented incidence matrix associated with the graph. Analogously, the nullity of the graph is the nullity of its oriented incidence matrix
Rank_(graph_theory)
MR 1964792, S2CID 34821155. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10
Strong_orientation
Quantified formulas with real-number variables
Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter M. (1993), Oriented Matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge:
Existential theory of the reals
Existential_theory_of_the_reals
American mathematician and academic administrator
Chapel Hill. Her dissertation, Affine Hyperplane Arrangements and Oriented Matroids, was supervised by Thomas H. Brylawski. She joined the University
Jenny_McNulty
Graph with group-labeled edges
network with gains, or generalized network, is connected with the frame matroid of the gain graph. Suppose we have some hyperplanes in R n given by equations
Gain_graph
Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented Matroids, Encyclopedia of Mathematics and Its Applications, vol. 46 (2nd ed
Bracket_ring
Vertices connected in pairs by edges
(simple) graph. Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a
Graph_(discrete_mathematics)
Matroid associated with a group
combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each
Dowling_geometry
Topics referred to by the same term
(graph theory), a symmetric tessellation of a closed surface Regular matroid, a matroid which can be represented over any field Regular paperfolding sequence
Regular
Tree which includes all vertices of a graph
also be expressed using the theory of matroids, according to which a spanning tree is a base of the graphic matroid, a fundamental cycle is the unique circuit
Spanning_tree
Affine subspace of a Euclidean space
example Dihedral angle (between two planes). See also Angles between flats.) Matroid Coplanarity Isometry Gallier, J. (2011). "Basics of Affine Geometry". Geometric
Flat_(geometry)
Trail in a graph that visits each edge once
bridgeless and almost-Eulerian), but they do not contain each other. Eulerian matroid, an abstract generalization of Eulerian graphs Five room puzzle Handshaking
Eulerian_path
Subfield of mathematical optimization
shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems. For NP-complete discrete optimization problems, current research
Combinatorial_optimization
Subspace of n-space whose dimension is (n-1)
which the concept of the dimension of a subspace is defined (see also Matroid § Hyperplanes (coatoms). The difference in dimension between a subspace
Hyperplane
American mathematician
Klee–Minty cube, the Browder–Minty theorem, the introduction of oriented regular matroids, and the Minty-Vitaver theorem on graph coloring. George Minty
George_J._Minty
minimums of finite collections of polynomials. Rota's basis conjecture: for matroids of rank n {\displaystyle n} with n {\displaystyle n} disjoint bases B i
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
mean matrices that are invertible over the field. Balanced matrix Regular matroid Special linear group Total dual integrality Hermite normal form The term
Unimodular_matrix
M. (1992), "8. Introduction to greedoids" (PDF), in White, Neil (ed.), Matroid Applications, Encyclopedia of Mathematics and its Applications, vol. 40
Rooted_graph
Measurement of graph sparsity
Westermann, H. H. (1992), "Forests, frames, and games: algorithms for matroid sums and applications", Algorithmica, 7 (1): 465–497, doi:10.1007/BF01758774
Degeneracy_(graph_theory)
the graphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In the bicircular matroid, a subset
Glossary_of_graph_theory
Branch of elementary mathematics
ISBN 978-4-431-54273-5. Koepf, Wolfram (2021). Computer Algebra: An Algorithm-Oriented Introduction. Springer Nature. ISBN 978-3-030-78017-3. Koetsier, Teun (2018)
Arithmetic
Field of knowledge
theory, geometry of numbers (method oriented), Diophantine analysis, and transcendence theory (problem oriented). Geometry is one of the oldest branches
Mathematics
Area of mathematics
A-not-B error. Further, since the middle of the 1990s cognitive science, oriented towards a system theoretical connectionism, has increasingly adopted the
Dynamical_systems_theory
Principle in mathematical optimization
of Max-Flow Min-Cut Theorem". Combinatorial Optimization: Networks and Matroids. Dover. pp. 117–120. ISBN 0-486-41453-1. Lemaréchal, Claude (2001). "Lagrangian
Duality_(optimization)
Embedding a graph in 3D space with no cycles interlinked
D. thesis, University of California, Berkeley. Truemper, Klaus (1992), Matroid Decomposition (PDF), Academic Press, pp. 100–101, archived from the original
Linkless_embedding
Area of geometry, about angles and lengths
Cohen; Lee B. Theodore; David Sklar (17 July 2009). Precalculus: A Problems-Oriented Approach, Enhanced Edition. Cengage Learning. ISBN 978-1-4390-4460-5. W
Trigonometry
Assignment of colors to edges of a graph
Westermann, Herbert H. (1992), "Forests, frames, and games: algorithms for matroid sums and applications", Algorithmica, 7 (5–6): 465–497, doi:10.1007/BF01758774
Edge_coloring
Type of graph in graph theory
"Exponentially many hypohamiltonian graphs", Graphs, Hypergraphs and Matroids III (Proc. Conf. Kalsk 1988), Zielona Góra: Higher College of Engineering
Hypohamiltonian_graph
Month in 1917
used by German armed forces during World War II, leading contributor to matroid and graph theory; as William Thomas Tutte, in Newmarket, Suffolk, England
May_1917
ORIENTED MATROID
ORIENTED MATROID
Boy/Male
British, English
Owner of a Rented Land
Girl/Female
Australian, French, Latin
Goal-oriented; Ambitious
Biblical
Oriental, Ancient, First
Boy/Male
Tamil
Lokeshwaran | லோகேஷà¯à®µà®°à®£
King of world is the single quote for this word. the person with this name would be more enchanting, Goal-oriented and would be able to adapt to any circumstances
Lokeshwaran | லோகேஷà¯à®µà®°à®£
Female
Hebrew
(לַיְלָה) Hebrew name LEILA means "night" or "dark Oriental beauty." Compare with other forms of Leila.
Boy/Male
American, British, English
From the Town on the High Ground; Owner of a Rented Estate
Boy/Male
Biblical
Oriental, ancient, first.
Girl/Female
Hindu, Indian, Tamil
Orient; Formerly
Female
Hebrew
 Variant spelling of Hebrew Leila, LEILAH means "night" or "dark Oriental beauty." Compare with another form of Leilah.
Boy/Male
Indian, Punjabi, Sikh
One who Lives God-oriented Life
Girl/Female
Hindu
Orient, Formerly
Boy/Male
Sikh
One who lives God oriented life
Boy/Male
Indian, Punjabi, Sikh
One who Lives God-oriented Live
Girl/Female
Hebrew American
Palm tree. Used as a symbolic oriental name due to the beauty and fruitfulness of the tree.
Girl/Female
Tamil
Purvika | பà¯à®°à¯à®µà®¿à®•à®¾
Orient, Formerly
Purvika | பà¯à®°à¯à®µà®¿à®•à®¾
Girl/Female
Tamil
Poorvika | பூரà¯à®µà®¿à®•à®¾Â
Orient, Formerly
Poorvika | பூரà¯à®µà®¿à®•à®¾Â
Boy/Male
Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Telugu
Printed or Written; Signet; Symbol; Female Version of Ankita; Stamped; Sign; Mark; Conquered
Boy/Male
Hindu
King of world is the single quote for this word. the person with this name would be more enchanting, Goal-oriented and would be able to adapt to any circumstances
Boy/Male
Gujarati, Hindu, Indian, Kannada, Punjabi, Sikh, Telugu
One who Lives a God Oriented Life
Boy/Male
Tamil
Harjeevan | ஹரஜீவநÂ
One who lives God oriented life
ORIENTED MATROID
ORIENTED MATROID
Boy/Male
Hindu, Indian
White; Washed; Pure; Clean
Boy/Male
Arabic
Servant of the Capable
Boy/Male
Sikh
Fighter of Love and truth
Girl/Female
Indian
Interrupter of the sacrifice of Daksha
Girl/Female
Spanish
Noble. Of the nobility.
Boy/Male
Tamil
Dharmansh | தரà¯à®®à®¾à®‚à®·
Surname or Lastname
English (West Midlands)
English (West Midlands) : elaborated form of Port.Dutch : from poort ‘gate’ + man ‘man’, an occupational name for a gatekeeper or a topographic name for someone who lived near the gates of a walled town (typically the man in charge of them). Compare Porter.American spelling of German Portmann.
Boy/Male
Afghan, Arabic, Muslim
Servant of the Most Merciful
Girl/Female
Muslim
Respectable
Surname or Lastname
English (Essex)
English (Essex) : probably a habitational name from either of two places called Binbrook. The one in Cambridge is named with Old English binnan ‘within’ + brÅc ‘brook’; the other, in Lincolnshire, is named with the Old English personal name Bynna + Old English brÅc.
ORIENTED MATROID
ORIENTED MATROID
ORIENTED MATROID
ORIENTED MATROID
ORIENTED MATROID
n.
A pearl of great luster.
imp. & p. p.
of Print
a.
Supplied with clients.
a.
Having friends;
a.
Tormented.
a.
Of or pertaining to the orient or east; eastern; concerned with the East or Orientalism; -- opposed to occidental; as, Oriental countries.
a.
Having three prongs; trident; tridentate; as, a tridented mace.
n.
The part of the horizon where the sun first appears in the morning; the east.
n.
Eastern Christians of the Greek rite.
v. t.
to render Oriental; to cause to conform to Oriental manners or conditions.
n.
A native or inhabitant of the Orient or some Eastern part of the world; an Asiatic.
imp. & p. p.
of Oint
imp. & p. p.
of Torment
v. t.
To define the position of, in relation to the orient or east; hence, to ascertain the bearings of.
imp. & p. p.
of Friend
v. t.
Fig.: To correct or set right by recurring to first principles; to arrange in order; to orientate.
imp. & p. p.
of Orientate
a.
Inclined to love; well-disposed.
a.
Eastern; oriental.
n.
The countries of Asia or the East.