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math-debrief
  • Math Debrief
  • Math: TIMELINE
  • 100-fundamentals
    • debrief-name: math section-code: 000 section-name: general section-desc: Elementary topics pervasive
      • About Mathematics
      • abstraction-in-math
      • About Math
      • Axiom schema
      • Basic concepts in math
      • Collections
      • Elementary concepts in objects
      • Elements of mathematics
      • math-as-a-language
      • Mathematical structures
      • List of mathematics-based methods
      • Mathematics and Reality
      • Mathematics: General
      • Controversial mathematics
      • the-elements-of-math
      • What is mathematics
    • The foundation of mathematics
      • Mathematical foundations
      • Foundations of Mathematics
      • Axiomatization of mathematics
      • Foundational crisis of mathematics
      • Foundations
      • Hilbert's problems
      • impl-of-math-in-set-theory
      • Gödel's Incompleteness Theorem
      • Theorems in the foundations of mathematics
      • The list of FOM candidates
      • Logicism
    • Philosophy of mathematics
      • Constructive mathematics
      • Constructive mathematics
      • Metamathematics
      • Philosophy of mathematics
      • Schools of mathematics
    • terms
      • Terms
      • Arithmetic
      • Axiom
      • The Axiomatic Method
      • discrete-math
      • 201 Discrete mathematics
      • Euclidean space
      • Formal system
      • Function
      • Generalization
      • Geometry
      • Higher-order
      • Impredicativity
      • Level of measurement
      • Mathematical definition
      • FAQ
      • Mathematical function
      • Mathematical induction
      • Mathematical object
      • Mathematical object
      • Equivalent definitions of mathematical structures
      • Mathematics
      • Mathematical model
      • mathematical-notation
      • Mathematical pages
      • Mathematical terminology
      • Mathematical adjective
      • Numbers
      • plane
      • Primer: Set Theory
      • Mathematical primitive
      • Set
      • Space
      • theory
      • Variable
  • 200 Set and Set theory
    • Sets: Hierarchy
    • set.TERMS
    • SETS › TOPICS
    • 201 Set concepts
      • Mathematical collections
      • The notion of sets
      • Specification of sets
    • Set cardinality
      • Cardinality of the continuum
      • Cardinality
      • Set Cardinality
      • cardinality2
      • Set cardinality
    • Set operations
      • Disjoint sets
      • Overlapping sets
      • Product
      • set-interactions
      • Set qualities
      • Set relations
    • Set properties
      • Basic set properties
      • Set properties
    • Set relations
      • Basic set relations
      • Disjoint sets
      • Inclusion relation
      • Membership Relation
      • Set membership
    • Summary
      • Set FAQ
      • Sets: Summary
    • Set theories
      • Axiomatic set theory
      • Set Theories
      • Naive Set Theory
      • Morse-Kelley set theory
      • von Neumann-Bernays-Gödel Set Theory
      • Quine's New Foundations (NF)
      • Cantor's set theory
      • Zermelo-Fraenkel set theory
    • Axioms of set theory
      • axiom-of-choice
      • The Axiom of Extensionality
      • Axiom of infinity
      • axiom-of-pairing
      • Axiom of powerset
      • Axiom of Regularity
      • Axiom of replacement
      • Axiom of union
      • Axiom of well-ordering
      • axiom-schema-of-comprehension
      • Axiom Schema of Specification
      • Axioms of set theories
      • List of axioms in set theory
      • ZFC Axioms
    • Sets: Terms
      • Bell Number
      • Cardinal number
      • Class (set theory)
      • Closure
      • empty-set
      • Extended set operations
      • Extensions by definitions
      • Family of sets
      • Fundamental sets
      • fundamental-sets2
      • Georg Cantor
      • History of set theory
      • Implementation of mathematics in set theory
      • Indexed family of sets
      • Extensional and intensional definitions
      • Involution
      • list-of-axioms-of-set-theory
      • Implementation of mathematics in set theory
      • Set membership
      • Naive Set Theory
      • Number of relations
      • empty-relation
      • Set Partitioning
      • Powerset
      • Russell's paradox
      • Set-builder notation
      • Set equivalence
      • Set Notation in latex
      • Set notation
      • Set partition
      • Intensional and extensional set specification
      • Set notation
      • Basic concepts in set theory
      • set-theory
      • Set Types
      • set
      • subset
      • Transfinite number
      • Tuples
      • ur-elements
  • Relations
    • basic-concepts
      • algebraic-axioms
      • Elements of a relation
      • Types of Relations
      • Named Relations
      • Relation theory
      • Relations
      • Types of relations
    • Relations
      • Definitions
      • Reflexivity
      • Symmetry
      • Transitivity
    • relation-properties
      • Uniqueness properties of relations
    • Types of relations
      • Transitivity
      • Binary Relation
      • Congruence relation
      • Connex relation
      • axioms-sets-zfc
      • Endorelation
      • Equivalence relation
      • Euclidean
      • Finitary relation
      • Heterogeneous relation
      • Homogeneous relation
      • Transitivity
      • Partial equivalence relation
      • Transitivity
      • Transitivity
      • Reflexive relation
      • Reflexivity
      • Index of relations
      • Serial relation
      • Symmetry
      • Transitivity
      • Ternary relation
      • Trichotomy
      • Universal relation
      • Well-foundedness
    • terms
      • Relations
      • Binary relation
      • Relations
      • _finitary-rel
      • Relations: Overview
      • Relations
      • Index of relations
      • Binary relations
      • Composition of relations
      • Equivalence class
      • Notation
      • Relation
      • Relations
      • Sets: Summary
      • Aggregation: Sets, Relations, Functions
  • Order theory
    • Order theory
    • List of order structures in mathematics
    • List of order theory topics
    • Order theory
      • Hasse diagram
      • Order theory
      • ordered-set
      • Partial order
      • Partially ordered set
      • Total order
  • Function Theory
    • Function Theory: GLOSSARY
    • Function Theory: HIERARCHY
    • Function Theory: LINKS
    • Function Theory: TERMS
    • Function Theory: TOPIC
    • Function Theory: WIKI
    • _articles
      • about-functions
      • Function
      • Formal definition
      • Definition
      • constant
      • Introduction
      • Types of functions
      • Functions: Summary of Notations
      • Functions: Overview
      • Properties of functions
      • Function properties
      • Functions: Summary
      • Function
    • Abjections
      • Bijective function
      • Function (abjections)
      • Injective function
      • Surjective function
    • topics
      • Function: TERMS
      • Codomain
      • Composition of functions
      • Currying
      • Division of functions
      • Domain
      • Function fixed points
      • Function cardinality
      • Function definition
      • Elements of a function
      • Function in mathematics
      • Function notion
      • Function operations
      • Function properties
      • Functional statements
      • Functions in programing languages
      • Image and Preimage
      • Image
      • Inverse function
      • Notion of functions
      • Number and types of functions between two sets
      • Operation
      • Range
      • Successor function
      • Time complexity classes
  • debrief-name: math section-code: 280 section-name: domain-theory section-desc:
    • Domain theory: LINKS
    • Domain theory
  • Logic
    • Logic: CHRONO TERMS
    • Logic: CLUSTERS
    • lo.GLOSSARY
    • Logic: Wiki links
    • 305-basic-concepts
      • Introduction to Logic
      • Argumentation
      • Logic: Basic terminology
      • Logic: Terminology
      • Truth function
      • Truth function
    • README
      • Mathematical Logic
      • Types of Logic
      • BHK interpretation
      • FOL
      • Index of Logic Forms
      • History of logic
      • Logic Indices
      • Interpretation of symbols in logic and math
      • logic-systems
      • Mathematical Logic: People and Events
      • Index of logical fallacies
      • Logical symbols
      • Mathematical conjecture
      • Mathematical induction
      • Mathematical lemma
      • Mathematical Logic
      • Mathematical proof
      • Mathematical theorem
      • Mathematical theory
      • Monotonicity of entailment
      • Satisfiability Modulo Theories
      • Sequent Calculus
      • Sequent
      • Tableaux
      • Truth tables
    • 360-propositional-logic
      • Propositional Logic
      • Propositional Logic
    • 370-predicate-logic
      • Predicate Logic
      • First-order logic
      • Predicate calculus
      • Examples of predicate formulae
    • 380-proof-theory
      • Argument-deduction-proof distinctions
      • Direct proof
      • Mathematical induction
      • Mathematical induction
      • Mathematical proof
      • Natural deduction
      • Natural deduction
      • Proof by induction
      • Proof by induction
      • proof-calculus
      • Proof Theory
      • Structural induction
      • System L
      • Proof theory
    • Logic: Indices
      • Gödel's Incompleteness Theorem
      • The History of Mathematical Logic
      • forallx
      • Logic for CS
      • Lectures in Logic and Set Theory
      • _logicomix
    • Logic
      • Logical connectives
      • Logical equivalence
    • Rules of Inference
      • WIKI
      • Conjunction elimination
      • Conjunction introduction
      • Cut rule
      • Disjunction elimination
      • Disjunction introduction
      • Disjunctive syllogism
      • Exportation
      • implication-elimination
      • implication-introduction
      • Rules of Inference: Index
      • Rules of inference
      • Rules of Inference for Natural Deduction
      • Logical Inference
      • Reiteration
      • Rule of inference
      • Structural rules
      • substitution
    • Logic
      • The principle of bivalence
      • The principle of explosion
      • The Law of Identity (ID)
      • Laws of thought
      • Properties of logic systems
      • List of laws in logic
      • The law of non-contradiction
    • Logic
      • Logic systems: LINKS
      • Logic system
      • logic-systems
      • logic-typ
      • logics-by-purpose
      • _logics
      • Affine logic
      • Algebraic logic
      • Bunched logic
      • Classical logic
      • Traditional first-order logic
      • Hoare logic
      • Linear logic
      • Modal logic
      • Non-monotonic logic
      • Syntax
      • Predicate logic
      • Propositional Logic
      • Relevance logic
      • Separation logic
      • Substructural logics
      • Syllogistic logic
    • Logic: Sections: Elementary
    • Logic: Topics
      • Pages in Logic
      • Logic ❱ Terms ❱ List
      • Logic ❱ Terms ❱ Definitions
      • Absoluteness
      • Assumption
      • Automated theorem proving
      • Canonical normal form
      • Categorical proposition
      • Classical linear logic
      • Consequence
      • Decidability
      • Deduction systems
      • deduction-theorem
      • Deductive reasoning
      • Diagonal lemma
      • Fallacy
      • Fitch notation
      • Formal language
      • formal-system
      • Formalism
      • Formula
      • functionally complete
      • Hilbert system
      • Hoare logic
      • horn-clause
      • Mathematical induction
      • Induction
      • Inductive Reasoning
      • Intuitionistic logic
      • Intuitionistic logic
      • Intuitionistic logic
      • Judgement
      • Judgments
      • Linear logic
      • Logic in computer science
      • Logic
      • Logical connective
      • Logical consequence
      • Logical constant
      • Logical form
      • axioms-sets
      • Logical reasoning
      • Ludics
      • Non-logical symbol
      • Predicate
      • Premise
      • Quantification
      • Realizability
      • Boolean satisfiability problem
      • DPLL algorithm
      • Satisfiability
      • Semantics of logic
      • Skolemization
      • SAT and SMT
      • Syntax
      • Tautology
      • Term
      • Unification
      • Validity
  • 510 Lambda Calculi
    • Lambda Calculus: GLOSSARY
    • Lambda calculi: LINKS
    • Lambda Calculus: OUTLINE
    • Lambda Calculus: Basic concepts
      • Introduction
      • Lambda expressions
      • Free variables
    • Lambda Calculi
      • Lambda calculus: LINKS
      • Lambda calculus combinators in Haskell
      • Lambda calculus: Combinators
      • Combinators
      • combos-all.js
      • combos-bird.js
      • combos-birds-list.js
      • combos-birds.js
      • Fixed-point combinator
      • Fixpoint operator
      • Lambda calculus: Fixpoint
    • combinatory-logic
      • algebraic-structures
      • Combinatory logic
      • Combinatory logic
      • relation-classification
      • 04-definition
    • Lambda calculus encoding schemes
      • bohm-berarducci-encoding
      • Index of Church encodings
      • Church encodings
      • Church Numerals
      • Encoding data structures
      • Encoding schemes in lambda calculi
      • Lambda encoding
      • Mogensen-Scott encoding
      • Parigot encoding
      • encodings
        • Encoding data structures
        • Encoding of Data Types in the λ-calculus
        • church-booleans
        • Church data structures
        • Church encoding
        • Church Numerals: Church encoding of natural numbers
        • Lambda Calculus: Church encoding
        • Lambda Calculus: Church encoding
        • church-numerals
        • Lambda Calculus: Church encoding: Numerals
        • Church pair
        • Pair
        • Lambda Calculus: Church encoding
        • Alternative encodings
        • Encoding schemes
        • Encoding schemes
        • Encodings in Untyped Lambda Calculus
        • Lambda calculus
        • Scott encoding
        • Lambda calculus: Scott encoding
    • lambda-calculus-evaluation
      • Call-by-name
      • Call-by-need
      • Call-by-value
    • lambda-calculus-forms
      • Beta normal form
      • Lambda terms
      • Fixity of lambda-terms
    • lambda-calculus-reductions
      • Alpha conversion
      • Beta reduction
      • Delta reduction
      • Eta conversion
      • Eta conversion
      • Lambda calculus: η-conversion
    • lambda-calculus
      • Alonzo Church
      • Inference rules for lambda calculus
      • Lambda Calculus: Introduction
      • Lambda abstraction
      • Lambda application
      • Lambda Calculus: Definition
      • About λ-calculus
      • Type inference
      • Lambda Calculus
      • Lambda Calculus: Introduction
      • Introduction to λ-calculus
      • Lambda calculus
      • Definition of Lambda Calculus
      • Functions in lambda calculus
      • History of Lambda Calculus
      • Using the Lambda Calculus
      • Name capturing
      • Variable occurrences
      • Variables
    • Lambda Calculus
      • Church-Rosser theorem
      • Curry's paradox
      • De Bruijn index
      • de Bruijn notation
      • Deductive lambda calculus
      • Kleene-Rosser paradox
      • Aspects of the lambda calculus
      • Function Refactoring
      • Lambda lifting
      • Let expression
      • Reduction strategy
      • Substitution
    • typed-lambda-calculi
      • Lambda Cube
      • Simply typed lambda calculus
      • System F
      • Typed lambda calculi
  • Type theory
    • Type Theory: GLOSSARY
    • Type theorists
    • Type Theory: SUMMARY
    • TERMS: Type Theory
      • Types
      • History of type theory
      • History of Type Theory
    • curry-howard-correspondence
      • The Curry-Howard Correspondence in Haskell
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard correspondence
      • Curry-Howard-Lambek correspondence - HaskellWiki
    • dependent-types
      • Dependent type
      • Dependent type
    • Hindley-Milner type-system
      • Hindley-Milner type system
      • Monomorphism vs polymorphism
      • Let-polymorphism
      • The Hindley-Milner type system
      • Algorithm W in Haskell
      • Hindley-Milner Type Inference: W Algorithm
      • hindley–milner-type-system
      • Hindley-Milner type system
      • HM inference examples
      • HM in ML
      • Type Inference
    • Homotopy type theory
      • Homotopy type theory
      • Univalent Type theory as the foundations of mathematics
    • Intuitionistic type theory
      • Inductive definition
      • Inductive type
      • Intuitionistic type theory
    • Type Theory
      • TTTools
      • Coinduction
      • Impredicativity
      • Lean
      • Subsumption
    • Type Theory : Topics
      • Type Theory : Terms
      • Recursion types
      • Recursive data type
      • Subtyping
      • Type Class
      • Type Equivalence
      • Type Inference
      • Type rule
      • Type system
      • Variance
    • type-theories
      • Calculus of Constructions
      • Constructive type theory
      • ramified-type-theory
      • simple-type-theory
      • Substructural type systems
    • type-theory-general
      • Linear types
      • History of Type Theory
      • Type Theory
      • Overview
      • Type Theory
  • Abstract Algebra
    • 410-group-theory
      • Abelian group
    • algebras
      • Associative Algebra
      • Field
      • Group-like algebraic structures
      • group
      • Lattice
      • Magma
      • monoid
      • Overview of Algebras
      • Quasigroup
      • Rack and quandle
      • Ring
      • Semigroup
      • Algebra of sets
      • Setoid
    • boolean-algebra
      • Boolean algebra
      • Axioms in Boolean Algebra
      • Boolean algebra
      • Boolean Algebra Laws
      • Boolean Algebra Laws
      • Two-element Boolean algebra
      • Boolean algebra
      • Boolean domain
    • terms
      • Algebra
      • Axioms of abstract algebra
      • Algebraic notation for algebraic data types
      • Algebraic structure
      • Algebraic structure
      • Field of sets
      • Homomorphism
      • Isomorphism
      • Algebraic structures
      • Mathematical structure
      • Polynomials
      • Relation algebra
  • Category Theory
    • CT GLOSSARY
    • Category Theory: OUTLINE
    • CT SUMMARY
    • A First Introduction to Categories (2009)
      • Sets, maps, composition
      • 02-history
      • axioms-logic
      • Bijection of functions
      • Commutative diagram
      • Directed graph
      • CT prerequisites
      • String diagram
      • Transitive closure
    • Category Theory Fundamentals
      • Introduction
      • Interpretation
      • Fundamental concepts
      • Category theory
      • Category
      • Category Theory: Definitions
    • Key concepts
      • Duality
      • Functor
      • Homeset
      • Initial Object
      • Morphism
      • Natural transformation
      • Object
      • Terminal Object
    • Categorical constructions
      • Categorical constructions
      • Coproduct
      • Diagram
      • Product
      • Universal construction
    • Types of categories
      • Concrete category
      • Discrete category
      • Functor category
      • Groupoid
      • Hask
      • Kleisli category
      • Locally small category
      • Monoid
      • monoidal-categories.md
      • Index of named categories
      • Opposite category
      • Ordered category
      • Set category
      • Small category
      • Subcategory
    • Types of Functors
      • Adjoint functor
      • relation-arity
      • Endofunctor
      • Faithful functor
      • Forgetful functor
      • Hom functor
      • Identity functor
      • Inverse functor
      • Monad
      • Powerset functor
    • Types of Morphisms
      • Anamorphism
      • Automorphism
      • Catamorphism
      • Endomorphism
      • Epimorphism
      • Homomorphism
      • Hylomorphism
      • Idempotent morphism
      • Identity morphism
      • Inverse morphism
      • Isomorphism
      • Metamorphism
      • monomorphism
      • Natural isomorphism
      • Split morphism
    • 20-advanced-concepts
      • Coalgebra
      • (Co)Inductive types
      • Recursion Schemes
    • Category Theory
      • Category Theory: TERMS
      • Algebraic Data Types
      • Category Theory
      • Category
      • Coproduct
      • Function type
      • Functoriality
      • Initial Object
      • Limits and Colimits
      • Natural Transformation
      • 5. Products
      • Terminal Object
    • Category Theory :: Contents
      • CT :: Links
      • Category Theory :: Terms
      • Category :: Definition
      • F-Algebra
      • Functor
      • Initial object
      • Monoid
      • Natural Transformation
      • Number of morphisms
      • Terminal object
      • Transitive closure
      • Types of morphisms
      • Categories by cardinality
      • Types of functors
  • Number Theory
    • Invariance and Monovariance Principle
    • 615-arithmetic
      • Addition
      • Aliquot sum
      • Arithmetic function
      • Laws
      • Arithmetic operations
      • Index of arithmetic operations
      • Arithmetic operations
      • Arithmetic
      • Divisibility rules
      • Divisibility
      • division
      • Divisor Function
      • Divisor Summatory Function
      • Divisor
      • Euclidean division
      • Hyperoperations
      • hyperops
      • Modular arithmetic
      • Multiplication
      • Number Theory: primer in numbers
      • Percentage
      • Rules of Divisibility
      • Subtraction
    • The fundamental sets of numbers
      • Algebraic numbers
      • Complex numbers
      • Fractions
      • Fundamental number sets
      • Imaginary numbers
      • Integers
      • Irrational numbers
      • Natural number
      • Rational numbers
      • Real numbers
      • Transcendental numbers
      • Ulam's spiral
      • The whole numbers
    • COUNTING THEORY
      • Counting Theory
      • counting
      • Fundamental Counting Rules
    • 630-combinatorics
      • Combinatorics
      • Combinations
      • Combinatorics
      • Counting theory
      • Counting theory
      • Enumerative combinatorics
      • Partition
      • Pascals triangle
      • Permutations
      • Twelvefold way
    • Probability theory
      • Statistics › Probability theory: Glossary
      • Statistics › Probability theory › Topics
      • Statistics › Probability theory › Wiki Links
      • Conditional Probability
      • Distribution
      • Probability theory
      • Probability
    • Number theory
      • euclids-lemma
      • gcd-lcm
      • Induction
      • Infinity
      • Numbers and numerals with interesting properties
      • Lagrange's four-square theorem
      • Matrix
      • Matrix
      • List of Number Systems
      • Number Theory
      • Number Theory with Glenn Olsen
      • Number
      • Arithmetic
      • Numbers
      • numeral-prefixes
      • Numeral system
      • Numeral
      • Ordinal numbers
      • Parity
      • Peano axioms
      • Polynomial
      • Polynomial
      • Positional notation
      • Probability
      • Symbol
      • Well Ordering Principle
    • topics
      • Coprimality
      • Facorization of composite numbers
      • Fundamental Theorem of Arithmetic
      • Prime factorisation
      • Prime number
      • Prime numbers
  • Theory of computation
    • Theory of computation: Abbreviations
    • Theory of computation: CHRONOLOGICAL TOPICS
    • Theory of computation: GLOSSARY
    • Theory of Computation: HIERARCHY
    • Theory of computation: LINKS
    • Theory of computation: TERMS
    • Theory of computation: TOPICS
    • Theory of computation: WIKI
    • Theory of Computation
      • _toc-more
      • Theory of Computation
    • 610-automata-theory
      • Abstract machine
      • Automata Theory
      • Automaton
      • Edit distance
      • Finite-state Machine
      • Automata Theory: WIKI
    • Formal systems
      • Abstract interpretation
      • Alphabet
      • Binary combinatory logic
      • Chomsky hierarchy
      • Epsilon calculus
      • Formal language
      • Iota and Jot
      • Regular expression
      • Regular Language
      • SKI combinator calculus
    • 621-grammar
      • Backus-Naur Form (BNF)
      • Context-free grammar
      • Context-sensitive grammar
      • Extended Backus–Naur Form (EBNF)
      • Regular Language
      • Terminal and nonterminal symbols
    • 622-syntax
      • Syntax
    • 624-semantics
      • Axiomatic semantics
      • Denotational Semantics: Summary
      • Denotational Semantics
      • Denotational Semantics
      • Denotational semantics
      • Formal semantics
      • Operational semantics
      • Semantics in CS
      • Semantics
    • 630-computability-theory
      • Computability (recursion) theory: TERMS
      • Computability (recursion) theory: TOPICS
      • Effective Computability
      • Church Thesis
      • Church-Turing Thesis
      • Computability theory
      • Computability
      • Computable function
      • Entscheidungsproblem
      • Halting problem
      • Machine that always halts
      • McCarthy Formalism
      • Super-recursive algorithm
      • Recursion theory
    • 632-recursive-function-theory
      • Recursion Theory
      • Ackermann function
      • General recursive function
      • Minimization operator
      • Partial functions
      • Recursion Function Theory
      • Sudan function
    • 634-primitive-recursive-functions
      • Primitive Recursive Function
      • Initial functions
      • The list of primitive recursive functions
      • Primitive combination
      • Primitive composition
      • Primitive recursion
      • Successor function
    • 640-models-of-computation
      • Models of computation: Summaries
      • Model of computation
    • 680-complexity-theory
      • Algorithmic Complexity
      • Complexity Theory
  • debrief-name: math section-code: 900 section-name: aggregations section-desc: Aggregations, indices,
    • Index of closures
    • List of mathematical entities
    • List of mathematical objects
    • Enumeration of mathematical structures
    • Math : Axioms as Formulae
    • 950-math-areas
      • Areas of mathematics
      • Areas of mathematics
    • 970-links
      • check
      • Math: Links
      • Math Debrief: Links
      • Math Primer: LINKS
      • Links
      • Math: LINKS: ncatlab
      • Math: LINKS
      • WIKI
      • WIKI
      • WIKI_ALL
      • Math: Wiki lists
      • Glossary of areas of mathematics
      • WIKI_collections
      • Mathematics for Computer Science
      • Mathematics Classification
      • math
      • Resources
      • Math on YouTubel Video Playlists
      • wiki resources
    • 980-hierarchy
      • HIERAR
      • Math: Hierarchy
      • Math HIERARCHY
      • classification
        • Mathematics
        • https://ncatlab.org/nlab/all_pages https://ncatlab.org/nlab/all_pages/reference https://ncatlab.org/
        • Math Classification and Topical Pages
        • Areas of mathematics
        • Areas of mathematics
        • Math Classification: CCS
        • Math hierarchy
        • Computational mathematics
        • Taxonomy: Mathematics
        • Areas of mathematics
        • Mathematics Subject Classification
        • Math fields
        • math-topics
        • Mathematics Subject Classification – MSC
        • MSC Classification Codes
        • mss-top-levels-filenames
        • MSC classification: Top Levels
        • Math classification
    • 990-appendix
      • Math glossary at ENCYCLOPÆDIA BRITANNICA
      • Bibliography
      • Math: Abbreviations
      • math.GLOSSARY
    • Math : Canon
      • Main branches of mathematics
      • Enumeration: Math paradigms
      • enum-math-symbols
      • List of mathematical theories
      • enum-algebras
        • Group-like algebraic structures
        • Group
        • Groupoid
        • magma
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  • Enumerative combinatorics
  • Analytic combinatorics
  • Partition theory
  • Graph theory
  • Design theory
  • Finite geometry
  • Order theory
  • Matroid theory
  • Extremal combinatorics
  • Probabilistic combinatorics
  • Algebraic combinatorics
  • Combinatorics on words
  • Geometric combinatorics
  • Topological combinatorics
  • Arithmetic combinatorics
  • Infinitary combinatorics
  • Combinatorial optimization
  • Coding theory
  • Discrete and computational geometry
  • Combinatorics and dynamical systems
  • Combinatorics and physics

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  1. Number Theory
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Combinatorics

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Fields of combinatorics

Enumerative combinatorics

Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a unified framework for counting permutations, combinations and partitions.

Analytic combinatorics

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae.

Partition theory

Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, and has connections with statistical mechanics.

Graph theory

Graphs are basic objects in combinatorics. The questions range from counting (e.g., the number of graphs on n vertices with k edges) to structural (e.g., which graphs contain Hamiltonian cycles) to algebraic questions (e.g., given a graph G and two numbers x and y, does the Tutte polynomial TG(x,y) have a combinatorial interpretation?). Although there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects. This is due to the fact that while combinatorial methods apply to many graph theory problems, the two are generally used to seek solutions to different problems.

Design theory

Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Block designs are combinatorial designs of a special type. This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850. The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups. The area has further connections to coding theory and geometric combinatorics.

Finite geometry

Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied. This area provides a rich source of examples for design theory. It should not be confused with discrete geometry (combinatorial geometry).

Order theory

Order theory is the study of partially ordered sets, both finite and infinite. Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras.

Matroid theory

Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation. Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part of order theory. It is now an independent field of study with a number of connections with other parts of combinatorics.

Extremal combinatorics

Extremal combinatorics studies extremal questions on set systems. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on 2n vertices is a complete bipartite graph Kn,n. Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate.

Ramsey theory is another part of extremal combinatorics. It states that any sufficiently large configuration will contain some sort of order. It is an advanced generalization of the pigeonhole principle.

Probabilistic combinatorics

In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random discrete object, such as a random graph? For instance, what is the average number of triangles in a random graph? Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find), simply by observing that the probability of randomly selecting an object with those properties is greater than 0. This approach (often referred to as the probabilistic method) proved highly effective in applications to extremal combinatorics and graph theory. A closely related area is the study of finite Markov chains, especially on combinatorial objects. Here again probabilistic tools are used to estimate the mixing time.

Often associated with Paul Erdős, who did the pioneering work on the subject, probabilistic combinatorics was traditionally viewed as a set of tools to study problems in other parts of combinatorics. However, with the growth of applications to analysis of algorithms in computer science, as well as classical probability, additive and probabilistic number theory, the area recently grew to become an independent field of combinatorics.

Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.

Combinatorics on words

Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory and probability. It has applications to enumerative combinatorics, fractal analysis, theoretical computer science, automata theory and linguistics. While many applications are new, the classical Chomsky–Schützenberger hierarchy of classes of formal grammars is perhaps the best-known result in the field.

Geometric combinatorics

Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension a convex polytope can have. Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. We should note that combinatorial geometry is an old fashioned name for discrete geometry.

Topological combinatorics

Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.

Arithmetic combinatorics

Arithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive number theory (sometimes also called additive combinatorics) refers to the special case when only the operations of addition and subtraction are involved. One important technique in arithmetic combinatorics is the ergodic theory of dynamical systems.

Infinitary combinatorics

Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part of set theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics.

Gian-Carlo Rota used the name continuous combinatorics to describe geometric probability, since there are many analogies between counting and measure.

Related fields

Combinatorial optimization

Combinatorial optimization is the study of optimization on discrete and combinatorial objects. It started as a part of combinatorics and graph theory, but is now viewed as a branch of applied mathematics and computer science, related to operations research, algorithm theory and computational complexity theory.

Coding theory

Coding theory started as a part of design theory with early combinatorial constructions of error-correcting codes. The main idea of the subject is to design efficient and reliable methods of data transmission. It is now a large field of study, part of information theory.

Discrete and computational geometry

Discrete geometry (also called combinatorial geometry) also began as a part of combinatorics, with early results on convex polytopes and kissing numbers. With the emergence of applications of discrete geometry to computational geometry, these two fields partially merged and became a separate field of study. There remain many connections with geometric and topological combinatorics, which themselves can be viewed as outgrowths of the early discrete geometry.

Combinatorics and dynamical systems

Combinatorial aspects of dynamical systems is another emerging field. Here dynamical systems can be defined on combinatorial objects. See for example graph dynamical system.

Combinatorics and physics

There are increasing interactions between combinatorics and physics, particularly statistical physics. Examples include an exact solution of the Ising model, and a connection between the Potts model on one hand, and the chromatic and Tutte polynomials on the other hand.

https://en.wikipedia.org/wiki/Combinatorics