Faster access than browser!
121 relations: Adjoint functors, Alexandrov topology, Alphabetical order, Antichain, Antisymmetric relation, Arithmetic, Bijection, Binary relation, Boolean algebra (structure), Boolean ring, Bounded set, Cartesian product, Category theory, Causal sets, Charles Sanders Peirce, Circle, Closed set, Closure operator, Complement (set theory), Complete lattice, Complete partial order, Completeness (order theory), Computer science, Continuous function, Cyclic order, Directed acyclic graph, Directed set, Disjoint union, Distributive lattice, Distributivity (order theory), Division (mathematics), Divisor, Domain theory, Duality (order theory), Ellipse, Ellipsis, Empty set, Equivalence of categories, Equivalence relation, Ernst Schröder, Filter (mathematics), Finite set, Function (mathematics), Function space, Functional analysis, Galois connection, Garrett Birkhoff, Genealogy, George Boole, Glossary of order theory, ..., Graph drawing, Graph theory, Greatest and least elements, Greatest common divisor, Hasse diagram, Heyting algebra, Hierarchy, Ideal (order theory), Idempotence, If and only if, Incidence algebra, Indifference curve, Infimum and supremum, Integer, Interval (mathematics), Interval order, Kolmogorov space, Lattice (order), Lawson topology, Least common multiple, Limit (category theory), Limit-preserving function (order theory), Lineal descendant, Mathematical logic, Mathematics, Maximal and minimal elements, Monotonic function, Natural number, Net (mathematics), Number, Numeral system, Open set, Order embedding, Order isomorphism, Partially ordered set, Pediatrics, Physician, Pointless topology, Pointwise, Power set, Preorder, Primary school, Product (category theory), Product order, Quantale, Rational number, Real number, Reflexive relation, Richard Dedekind, Scott continuity, Semiorder, Set (mathematics), Set theory, Specialization (pre)order, Stone duality, Subbase, Subset, Subtraction, Surjective function, Topology, Total order, Transitive relation, Union (set theory), Universal algebra, Upper and lower bounds, Upper topology, Vertex (graph theory), Weak ordering, Well-order, Well-quasi-ordering, Zorn's lemma. Expand index (71 more) »
Adjoint functors
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
New!!: Order theory and Adjoint functors · See more »
Alexandrov topology
In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open.
New!!: Order theory and Alexandrov topology · See more »
Alphabetical order
Alphabetical order is a system whereby strings of characters are placed in order based on the position of the characters in the conventional ordering of an alphabet.
New!!: Order theory and Alphabetical order · See more »
Antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
New!!: Order theory and Antichain · See more »
Antisymmetric relation
In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other.
New!!: Order theory and Antisymmetric relation · See more »
Arithmetic
Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.
New!!: Order theory and Arithmetic · See more »
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
New!!: Order theory and Bijection · See more »
Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
New!!: Order theory and Binary relation · See more »
Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
New!!: Order theory and Boolean algebra (structure) · See more »
Boolean ring
In mathematics, a Boolean ring R is a ring for which x2.
New!!: Order theory and Boolean ring · See more »
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.
New!!: Order theory and Bounded set · See more »
Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
New!!: Order theory and Cartesian product · See more »
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
New!!: Order theory and Category theory · See more »
Causal sets
The causal sets program is an approach to quantum gravity.
New!!: Order theory and Causal sets · See more »
Charles Sanders Peirce
Charles Sanders Peirce ("purse"; 10 September 1839 – 19 April 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".
New!!: Order theory and Charles Sanders Peirce · See more »
Circle
A circle is a simple closed shape.
New!!: Order theory and Circle · See more »
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
New!!: Order theory and Closed set · See more »
Closure operator
In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).
New!!: Order theory and Closure operator · See more »
Complement (set theory)
In set theory, the complement of a set refers to elements not in.
New!!: Order theory and Complement (set theory) · See more »
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
New!!: Order theory and Complete lattice · See more »
Complete partial order
In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties.
New!!: Order theory and Complete partial order · See more »
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset).
New!!: Order theory and Completeness (order theory) · See more »
Computer science
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.
New!!: Order theory and Computer science · See more »
Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
New!!: Order theory and Continuous function · See more »
Cyclic order
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.
New!!: Order theory and Cyclic order · See more »
Directed acyclic graph
In mathematics and computer science, a directed acyclic graph (DAG), is a finite directed graph with no directed cycles.
New!!: Order theory and Directed acyclic graph · See more »
Directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.
New!!: Order theory and Directed set · See more »
Disjoint union
In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.
New!!: Order theory and Disjoint union · See more »
Distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.
New!!: Order theory and Distributive lattice · See more »
Distributivity (order theory)
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima.
New!!: Order theory and Distributivity (order theory) · See more »
Division (mathematics)
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.
New!!: Order theory and Division (mathematics) · See more »
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
New!!: Order theory and Divisor · See more »
Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains.
New!!: Order theory and Domain theory · See more »
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd.
New!!: Order theory and Duality (order theory) · See more »
Ellipse
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.
New!!: Order theory and Ellipse · See more »
Ellipsis
An ellipsis (plural ellipses; from the ἔλλειψις, élleipsis, 'omission' or 'falling short') is a series of dots (typically three, such as "…") that usually indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning.
New!!: Order theory and Ellipsis · See more »
Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
New!!: Order theory and Empty set · See more »
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
New!!: Order theory and Equivalence of categories · See more »
Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
New!!: Order theory and Equivalence relation · See more »
Ernst Schröder
Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic.
New!!: Order theory and Ernst Schröder · See more »
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set.
New!!: Order theory and Filter (mathematics) · See more »
Finite set
In mathematics, a finite set is a set that has a finite number of elements.
New!!: Order theory and Finite set · See more »
Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
New!!: Order theory and Function (mathematics) · See more »
Function space
In mathematics, a function space is a set of functions between two fixed sets.
New!!: Order theory and Function space · See more »
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
New!!: Order theory and Functional analysis · See more »
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).
New!!: Order theory and Galois connection · See more »
Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician.
New!!: Order theory and Garrett Birkhoff · See more »
Genealogy
Genealogy (from γενεαλογία from γενεά, "generation" and λόγος, "knowledge"), also known as family history, is the study of families and the tracing of their lineages and history.
New!!: Order theory and Genealogy · See more »
George Boole
George Boole (2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland.
New!!: Order theory and George Boole · See more »
Glossary of order theory
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory.
New!!: Order theory and Glossary of order theory · See more »
Graph drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.
New!!: Order theory and Graph drawing · See more »
Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
New!!: Order theory and Graph theory · See more »
Greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
New!!: Order theory and Greatest and least elements · See more »
Greatest common divisor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
New!!: Order theory and Greatest common divisor · See more »
Hasse diagram
In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
New!!: Order theory and Hasse diagram · See more »
Heyting algebra
In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.
New!!: Order theory and Heyting algebra · See more »
Hierarchy
A hierarchy (from the Greek hierarchia, "rule of a high priest", from hierarkhes, "leader of sacred rites") is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "above", "below", or "at the same level as" one another A hierarchy can link entities either directly or indirectly, and either vertically or diagonally.
New!!: Order theory and Hierarchy · See more »
Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset).
New!!: Order theory and Ideal (order theory) · See more »
Idempotence
Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.
New!!: Order theory and Idempotence · See more »
If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
New!!: Order theory and If and only if · See more »
Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity.
New!!: Order theory and Incidence algebra · See more »
Indifference curve
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent.
New!!: Order theory and Indifference curve · See more »
Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
New!!: Order theory and Infimum and supremum · See more »
Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
New!!: Order theory and Integer · See more »
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
New!!: Order theory and Interval (mathematics) · See more »
Interval order
In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2.
New!!: Order theory and Interval order · See more »
Kolmogorov space
In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other.
New!!: Order theory and Kolmogorov space · See more »
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.
New!!: Order theory and Lattice (order) · See more »
Lawson topology
In mathematics and theoretical computer science the Lawson topology, named after J. D. Lawson, is a topology on partially ordered sets used in the study of domain theory.
New!!: Order theory and Lawson topology · See more »
Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.
New!!: Order theory and Least common multiple · See more »
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
New!!: Order theory and Limit (category theory) · See more »
Limit-preserving function (order theory)
In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima.
New!!: Order theory and Limit-preserving function (order theory) · See more »
Lineal descendant
A lineal descendant, in legal usage, is a blood relative in the direct line of descent – the children, grandchildren, great-grandchildren, etc.
New!!: Order theory and Lineal descendant · See more »
Mathematical logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
New!!: Order theory and Mathematical logic · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Order theory and Mathematics · See more »
Maximal and minimal elements
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.
New!!: Order theory and Maximal and minimal elements · See more »
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
New!!: Order theory and Monotonic function · See more »
Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
New!!: Order theory and Natural number · See more »
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
New!!: Order theory and Net (mathematics) · See more »
Number
A number is a mathematical object used to count, measure and also label.
New!!: Order theory and Number · See more »
Numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
New!!: Order theory and Numeral system · See more »
Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
New!!: Order theory and Open set · See more »
Order embedding
In mathematical order theory, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another.
New!!: Order theory and Order embedding · See more »
Order isomorphism
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets).
New!!: Order theory and Order isomorphism · See more »
Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
New!!: Order theory and Partially ordered set · See more »
Pediatrics
Pediatrics (also spelled paediatrics or pædiatrics) is the branch of medicine that involves the medical care of infants, children, and adolescents.
New!!: Order theory and Pediatrics · See more »
Physician
A physician, medical practitioner, medical doctor, or simply doctor is a professional who practises medicine, which is concerned with promoting, maintaining, or restoring health through the study, diagnosis, and treatment of disease, injury, and other physical and mental impairments.
New!!: Order theory and Physician · See more »
Pointless topology
In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points.
New!!: Order theory and Pointless topology · See more »
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
New!!: Order theory and Pointwise · See more »
Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
New!!: Order theory and Power set · See more »
Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.
New!!: Order theory and Preorder · See more »
Primary school
A primary school (or elementary school in American English and often in Canadian English) is a school in which children receive primary or elementary education from the age of about seven to twelve, coming after preschool, infant school and before secondary school.
New!!: Order theory and Primary school · See more »
Product (category theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
New!!: Order theory and Product (category theory) · See more »
Product order
In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product.
New!!: Order theory and Product order · See more »
Quantale
In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras).
New!!: Order theory and Quantale · See more »
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
New!!: Order theory and Rational number · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Order theory and Real number · See more »
Reflexive relation
In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself.
New!!: Order theory and Reflexive relation · See more »
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.
New!!: Order theory and Richard Dedekind · See more »
Scott continuity
In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.
New!!: Order theory and Scott continuity · See more »
Semiorder
In order theory, a branch of mathematics, a semiorder is a type of ordering that may be determined for a set of items with numerical scores by declaring two items to be incomparable when their scores are within a given margin of error of each other, and by using the numerical comparison of their scores when those scores are sufficiently far apart.
New!!: Order theory and Semiorder · See more »
Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
New!!: Order theory and Set (mathematics) · See more »
Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
New!!: Order theory and Set theory · See more »
Specialization (pre)order
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.
New!!: Order theory and Specialization (pre)order · See more »
Stone duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets.
New!!: Order theory and Stone duality · See more »
Subbase
In topology, a subbase (or subbasis) for a topological space with topology is a subcollection of that generates, in the sense that is the smallest topology containing.
New!!: Order theory and Subbase · See more »
Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
New!!: Order theory and Subset · See more »
Subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.
New!!: Order theory and Subtraction · See more »
Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
New!!: Order theory and Surjective function · See more »
Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
New!!: Order theory and Topology · See more »
Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
New!!: Order theory and Total order · See more »
Transitive relation
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
New!!: Order theory and Transitive relation · See more »
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
New!!: Order theory and Union (set theory) · See more »
Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
New!!: Order theory and Universal algebra · See more »
Upper and lower bounds
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.
New!!: Order theory and Upper and lower bounds · See more »
Upper topology
In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton \ is the order section a.
New!!: Order theory and Upper topology · See more »
Vertex (graph theory)
In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).
New!!: Order theory and Vertex (graph theory) · See more »
Weak ordering
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other.
New!!: Order theory and Weak ordering · See more »
Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
New!!: Order theory and Well-order · See more »
Well-quasi-ordering
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, … from X contains an increasing pair x_i\le x_j with i.
New!!: Order theory and Well-quasi-ordering · See more »
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
New!!: Order theory and Zorn's lemma · See more »
Redirects here:
Order relation, Order structure.
References
[1] https://en.wikipedia.org/wiki/Order_theory
