Order (mathematics) explained

Order in mathematics may refer to:

Set theory

• Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary
• Ordered set
• Order in Ramsey theory, uniform structures in consequence to critical set cardinality

Algebra

• Order (group theory), the cardinality of a group or period of an element
• Order of a polynomial (disambiguation)
• Order of a square matrix, its dimension
• Order (ring theory), an algebraic structure
• Ordered group
• Ordered field

Analysis

• Order (differential equation) or order of highest derivative, of a differential equation
• Leading-order terms
• NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline
• Order of convergence, a measurement of convergence
• Order of derivation
• Order of an entire function
• Order of a power series, the lowest degree of its terms
• Ordered list, a sequence or tuple
• Orders of approximation in Big O notation
• Z-order (curve), a space-filling curve

Arithmetic

• Multiplicative order in modular arithmetic
• Order of operations
• Orders of magnitude, a class of scale or magnitude of any amount

Combinatorics

• Order in the Josephus permutation
• Ordered selections and partitions of the twelvefold way in combinatorics
• Ordered set, a bijection, cyclic order, or permutation
• Unordered subset or combination
• Weak order of permutations

Fractals

• Complexor, or complex order in fractals
• Order of extension in Lakes of Wada
• Order of fractal dimension (Rényi dimensions)
• Orders of construction in the Pythagoras tree

Geometry

• Long-range aperiodic order, in pinwheel tiling, for instance

Graphs

• Graph order, the number of nodes in a graph
• First order and second order logic of graphs
• Topological ordering of directed acyclic graphs
• Degeneracy ordering of undirected graphs
• Elimination ordering of chordal graphs
• Order, the complexity of a structure within a graph: see haven (graph theory) and bramble (graph theory)

Logic

In logic, model theory and type theory:

• Zeroth-order logic
• First-order logic
• Second-order logic
• Higher-order logic

Order theory

• Order (journal), an academic journal on order theory
• Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order
• Glossary of order theory
• Lexicographical order, an ordering method on sequences analogous to alphabetical order on words
• List of order topics, list of order theory topics
• Order theory, study of various binary relations known as orders
• Order topology, a topology of total order for totally ordered sets
• Ordinal numbers, numbers assigned to sets based on their set-theoretic order
• Partial order, often called just "order" in order theory texts, a transitive antisymmetric relation
• Total order, a partial order that is also total, in that either the relation or its inverse holds between any unequal elements

Statistics

• Order statistics
• First-order statistics, e.g., arithmetic mean, median, quantiles
• Second-order statistics, e.g., correlation, power spectrum, variance
• Higher-order statistics, e.g., bispectrum, kurtosis, skewness