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Associative algebra used in combinatorics
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative
Incidence_algebra
with unity. Category algebras generalize the notions of group algebras and incidence algebras, just as categories generalize the notions of groups and partially
Category_algebra
Topics referred to by the same term
welfare Incidence algebra, associative algebra used in combinatorics, a branch of mathematics Incidence geometry, the study of relations of incidence between
Incidence
Hopcroft's problem of finding point–line incidences Incidence algebra Incidence geometry Incidence matrix Incidence structure Levi graph Menelaus theorem
Incidence_(geometry)
Multiplicative function in number theory
partially ordered set (poset) is assigned an incidence algebra. One distinguished member of this algebra is that poset's "Möbius function". The classical
Möbius_function
group Hall algebra Hecke algebra of a locally compact group Heyting algebra Hopf algebra Hurwitz algebra Hypercomplex algebra Incidence algebra Iwahori–Hecke
List_of_algebras
Relation between pairs of arithmetic functions
applying to the set of the natural numbers ordered by divisibility: see incidence algebra. The classic version states that if g and f are arithmetic functions
Möbius_inversion_formula
tree languages and tree automata, etc. Group algebra (disambiguation) Incidence algebra Path algebra McNulty & Shallon 1983, pp. 206–231. Davey et al
Graph_algebra
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
Given a locally finite poset P we can define its incidence algebra. Elements of the incidence algebra are functions ƒ that assign to each interval [x, y]
Locally_finite_poset
Directed graph which is also a multigraph
theory Graph algebra Group ring Incidence algebra Quiver diagram Semi-invariant of a quiver Toric variety Derived noncommutative algebraic geometry - Quivers
Quiver_(mathematics)
Index of lists with the same name
function Witten zeta function of a Lie group Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value
List_of_zeta_functions
Topological algebra associated to continuous groups
Graph algebra Incidence algebra Hecke algebra of a locally compact group Path algebra Groupoid algebra Stereotype algebra Stereotype group algebra Hopf
Group algebra of a locally compact group
Group_algebra_of_a_locally_compact_group
Journal publisher
Topology Innovations in Incidence Geometry—Algebraic, Topological and Combinatorial Involve: A Journal of Mathematics Journal of Algebraic Statistics Journal
Mathematical Sciences Publishers
Mathematical_Sciences_Publishers
Set of finitely supported functions from a group to a ring
to the monoid ring and thence to the category algebra, of which another example is the incidence algebra. If a group has a length function – for example
Group_ring
Branch of mathematics
of Stone duality. Causal sets Cyclic order Hierarchy (mathematics) Incidence algebra Order theory glossary Bertrand Russell (1901) Mind 10(2) Immanuel
Order_theory
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
The Kronecker delta forms the multiplicative identity element of an incidence algebra. The Kronecker delta is an elementary recursive function. In probability
Kronecker_delta
Glossary of terms used in branch of mathematics
The dual notion is called filter. Incidence algebra. The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals
Glossary_of_order_theory
Ring that is also a vector space or a module
[citation needed] The Weyl algebra An Azumaya algebra The Clifford algebras, which are useful in geometry and physics. Incidence algebras of locally finite partially
Associative_algebra
Field of mathematics which studies incidence structures
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that
Incidence_geometry
Poset representing certain properties of a polytope
underlying abstract polytope, a purely algebraic partially ordered set which captures the pattern of connections (or incidences) between the various structural
Abstract_polytope
Conjecture on zeros of the zeta function
o(n^{1/2+\epsilon })} for all ϵ > 0 {\displaystyle \epsilon >0} (see incidence algebra). The Riemann hypothesis is equivalent to many other conjectures about
Riemann_hypothesis
Generalization of the concept of category that allows morphisms of multiple arity
triangles in a multiorder allows one to construct an associative algebra called its incidence algebra. Any element that is nonzero on all unit arrows has a compositional
Multicategory
Mathematical set with an ordering
poset – Partially ordered set equipped with a rank function Incidence algebra – Associative algebra used in combinatorics Lattice – Set whose pairs have minima
Partially_ordered_set
order Weak order of permutations Bruhat order on a Coxeter group Incidence algebra Monotonic Pointwise order of functions Galois connection Order embedding
List_of_order_theory_topics
Overview of and topical guide to combinatorics
formula Parity, even and odd permutations Combinatorial Nullstellensatz Incidence algebra Greedy algorithm Divide and conquer algorithm Akra–Bazzi method Dynamic
Outline_of_combinatorics
Topological invariant in mathematics
the integer μ(0,1), where μ is the Möbius function in that poset's incidence algebra. This can be further generalized by defining a rational valued Euler
Euler_characteristic
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is
Geometric_algebra
projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly
Intersection_theorem
Mathematical operation on arithmetical functions
convolution is a special case of the convolution multiplication for the incidence algebra of a poset, in this case the poset of positive integers ordered by
Dirichlet_convolution
Topics referred to by the same term
arguments and zero otherwise, and which forms the identity element of an incidence algebra Modular discriminant (Δ), a complex function in Weierstrass's elliptic
Delta function (disambiguation)
Delta_function_(disambiguation)
Computer system for solving algebra problems
theory Algebraic geometry Arithmetic geometry Finite incidence structures Cryptography Coding theory Optimization Comparison of computer algebra systems
Magma (computer algebra system)
Magma_(computer_algebra_system)
Counting technique in combinatorics
numbers. Therefore, (2) is seen as the Möbius inversion formula for the incidence algebra of the partially ordered set of all subsets of A. For a generalization
Inclusion–exclusion_principle
Matrix that shows the relationship between two classes of objects
mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If
Incidence_matrix
Italian-American mathematician (1932–1999)
location of the zeroes of polynomials. He worked on the theory of incidence algebras (which generalize the 19th-century theory of Möbius inversion) and
Gian-Carlo_Rota
Branch of mathematics
of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a
Geometry
Geometric concept of a 2D space with "points at infinity" adjoined
planar ternary ring. Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane
Projective_plane
Number of intersection points of algebraic curves and hypersurfaces
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form
Bézout's_theorem
Mirror-like wave reflection
equivalently expressed using linear algebra. The direction of a reflected ray is determined by the vector of incidence and the surface normal vector. Given
Specular_reflection
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching, is totally unimodular (TU). (The unoriented incidence matrix
Unimodular_matrix
Branch of mathematics
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Aspect of geometry
the related flag concept from linear algebra. A flag is maximal if it is not contained in a larger flag. An incidence geometry (Ω, I) has rank r if Ω can
Flag_(geometry)
Discrete analog of a derivative
algebras. Difference operator generalizes to Möbius inversion over a partially ordered set. As a convolution operator: Via the formalism of incidence
Finite_difference
Type of geometry
geometry Fundamental theorem of projective geometry Grassmann–Cayley algebra Incidence (mathematics) Projective line Projective line over a ring Projective
Projective_geometry
Branch of algebraic geometry
mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered
Arithmetic_geometry
Characteristic of undirected graphs
Schochetman, Irwin E. (1995), "On the minors of an incidence matrix and its Smith normal form", Linear Algebra and Its Applications, 218: 213–224, doi:10
Rank_(graph_theory)
Type of polygon
to all associative, non-commutative division algebras; over the reals these are limited to the algebra of quaternions, which has degree 2 (and dimension
Moufang_polygon
Geometry with 7 points and 7 lines
finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries
Fano_plane
Structure dual to a unital associative algebra
sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams
Coalgebra
Area of discrete mathematics
where he drew an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams. The definition of a graph can vary, but one can
Graph_theory
called "varieties", with a symmetric, reflexive relation on X called "incidence", together with a function τ called the "type map" from X to a set Δ whose
Buekenhout_geometry
Geometry without using coordinates
showed that algebraic axioms, such as commutativity and associativity of addition and multiplication, were in fact consequences of incidence of lines in
Synthetic_geometry
Any collection of sets, or subsets of a set
simplicial complex. An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called the incidence relation, specifying
Family_of_sets
Completion of the usual space with "points at infinity"
definition, which is more often encountered in modern textbooks. Using linear algebra, a projective space of dimension n is defined as the set of the vector
Projective_space
In universal algebra, a variety of algebras means the class of all algebraic structures of a given signature satisfying a given set of identities. One
Locally_finite_variety
Generalised concept of incidence structure of polygons
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective
Generalized_polygon
Branch of mathematics
operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras;
Noncommutative_geometry
1155/S1073792801000198, MR 1827084. Szczesny, M. (2010), Pre-Lie algebras and incidence categories of colored rooted trees, vol. 1007, p. 4784, arXiv:1007
Pre-Lie_algebra
Mathematics textbook on algebraic combinatorics
ordered sets and lattices, including material on Möbius functions of incidence algebras, Sperner's theorem on antichains in power sets, special classes of
Combinatorics:_The_Rota_Way
1957 book by Emil Artin
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley
Geometric_Algebra_(book)
Observation concerning cancer rates
species level, the incidence of cancer does not appear to correlate with the number of cells in an organism. For example, the incidence of cancer in humans
Peto's_paradox
Field of mathematics dealing with three-dimensional Euclidean spaces
of its radius. Basic topics in solid geometry and stereometry include: incidence of planes and lines dihedral angle and solid angle the cube, cuboid, parallelepiped
Solid_geometry
Equations of light transmission and reflection
normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers
Fresnel_equations
notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures
List_of_theorems
Branch of geometry that studies combinatorial properties and constructive methods
polyhedra Incidence structures generalize planes (such as affine, projective, and Möbius planes) as can be seen from their axiomatic definitions. Incidence structures
Discrete_geometry
Geometric system with a finite number of points
ISBN 978-1107518438. Weisstein, Eric W. "finite geometry". MathWorld. Incidence Geometry by Eric Moorhouse Algebraic Combinatorial Geometry by Terence Tao Essay on Finite
Finite_geometry
Space with one dimension
In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality
One-dimensional_space
postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. Occasionally named modern algebra in course titles
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Set of n^3 + 1 points arranged into subsets of n + 1
(1966), "Some remarks concerning the Ree group of type (G2)", Journal of Algebra, 3 (2): 256–259, doi:10.1016/0021-8693(66)90014-7 Penttila, T.; Royle,
Unital_(geometry)
Topics referred to by the same term
folded up to form a polyhedron An incidence structure consisting of points and parallel classes of lines An operator algebra in local quantum field theory
Net
Branch of mathematics
manifolds. It uses the techniques of vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry
Differential_geometry
Type of measurement in quantum mechanics
Marcus; Flammia, Steven; McConnell, Gary; Yard, Jon (2017-04-24). "SICs and Algebraic Number Theory". Foundations of Physics. 47 (8): 1042–1059. arXiv:1701
SIC-POVM
Mathematician and engineer
"Ordered Incidence Geometry" (PDF). GI-LECTURE-5.dvi. Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory
Adi_Ben-Israel
Concept in quantum information theory
1088/0305-4470/14/12/019. Planat, M.; et al. (14 November 2006). "A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements"
Mutually_unbiased_bases
API for graph data and graph operations
linear algebra. GraphBLAS is built upon the notion that a sparse matrix can be used to represent graphs as either an adjacency matrix or an incidence matrix
GraphBLAS
In projective geometry, a bijection between projective spaces that preserves collinearity
treated differently. For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between
Collineation
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
These arguments rely on results from algebraic geometry bounding the number of incidences between various algebraic curves. The technique of polynomial
Polynomial method in combinatorics
Polynomial_method_in_combinatorics
American astrophysicist (born 1967)
served in the US Navy until 1986. He credits the Navy with teaching him algebra. He left the Navy with an honorable discharge due to a skin condition from
Hakeem_Oluseyi
General concept and operation in mathematics
object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces
Duality_(mathematics)
Mathematical model of the physical space
dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two
Euclidean_geometry
Branch of computer science
Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry Concepts Features Dimension Straightedge
Computational_geometry
Mathematics of varieties with integer coordinates
polynomial equations) by means of powerful methods in algebraic geometry. The extensive development of algebraic geometry in the 20th century produced powerful
Diophantine_geometry
Belgian mathematician (1930–2021)
was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits
Jacques_Tits
Relationship between elements of two sets
Correspondence (algebraic geometry), a binary relation defined by algebraic equations Hasse diagram, a graphic means to display an order relation Incidence structure
Binary_relation
Algorithmic problem on point-line incidence
of the lines. More generally, one may ask for the number of point–line incidences. Both versions of the problem can be solved in time O ( n 4 / 3 ) {\displaystyle
Hopcroft's_problem
Problem of determining if a Boolean formula could be made true
TRUE just when exactly one of its arguments is. Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent
Boolean satisfiability problem
Boolean_satisfiability_problem
Construction in group theory
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL)
Projective_linear_group
Generalization of graph theory
ISSN 0012-365X. Parui, Samiron (2025). "On the incidence matrices of hypergraphs". Linear and Multilinear Algebra. 73 (17): 3861–3880. arXiv:2409.16055. doi:10
Hypergraph
Concept in projective geometry
subspaces of dimension d − k − 1, reversing inclusion and preserving incidence. Correlations are also called reciprocities or reciprocal transformations
Correlation (projective geometry)
Correlation_(projective_geometry)
Property of a mathematical space
cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension. For the non-free
Dimension
On the number of spanning trees in a graph
other entries in column k are 0 (see oriented incidence matrix for understanding this modified incidence matrix E). For the preceding example (with n =
Kirchhoff's_theorem
Complete reflection of a wave
neither a name nor an algebraic expression for the critical angle, he gave numerical examples for glass-to-air and water-to-air incidence, noted the large
Total_internal_reflection
Common elements of two or more sets
geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower
Intersection
Matrix representation of a graph
has been used. The | v | × | e | {\textstyle |v|\times |e|} oriented incidence matrix B with element Bve for the vertex v and the edge e (connecting
Laplacian_matrix
matroid theory the nullity of the graph is the nullity of the oriented incidence matrix M associated with the graph. The nullity of M is given by m − n
Nullity_(graph_theory)
respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph. Let G := ( V , E ) {\displaystyle G:=(V,E)} be a
Edge_and_vertex_spaces
Bound on the number of incidences between points and lines in the plane
Tao obtained near sharp upper bounds for the number of incidences between points and algebraic varieties in higher dimensions, when the points and varieties
Szemerédi–Trotter_theorem
Fundamental object of geometry
structure (algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions or an algebra of sets
Point_(geometry)
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