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Class of mathematical orderings
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset
Well-order
Generalized alphabetical order
lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences
Lexicographic_order
Statement that all non empty subsets of positive numbers contains a least element
In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set
Well-ordering_principle
Theorem that every set can be well-ordered
the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if
Well-ordering_theorem
Mathematical concept for comparing objects
In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which
Well-quasi-ordering
Type of binary relation
In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order, then
Well-founded_relation
Order whose elements are all comparable
mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤
Total_order
Generalization of "n-th" to infinite cases
linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered
Ordinal_number
Mathematician (1845–1918)
natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines
Georg_Cantor
Order-preserving mathematical function
or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus,
Monotonic_function
Isomorphism type of ordered sets
∗ {\displaystyle \sigma ^{*}} . The order type of a well-ordered set X is sometimes expressed as ord(X). The order type of the integers and rationals is
Order_type
Order for the terms of a polynomial
order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent
Monomial_order
Branch of mathematics
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing
Order_theory
Subset of incomparable elements
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
Antichain
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.
Partially_ordered_set
Reflexive and transitive binary relation
In mathematics, in particular in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant
Preorder
In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two
Series-parallel_partial_order
denoted property (K) Well-founded relation Ordinal number Well-quasi-ordering Semilattice Lattice (Directed) complete partial order, (d)cpo Bounded complete
List_of_order_theory_topics
Certain topology in mathematics
is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given
Order_topology
Nonempty, upper-bounded, downward-closed subset
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Term in the mathematical area of 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
Duality_(order_theory)
British order of chivalry established in 1917
The Most Excellent Order of the British Empire is a British order of chivalry, rewarding valuable service in a wide range of useful activities. It comprises
Order_of_the_British_Empire
Glossary of terms used in branch of mathematics
related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources
Glossary_of_order_theory
Existence of certain infima or suprema of a given poset
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset)
Completeness_(order_theory)
Mathematical concept
principle is also true for arbitrary well-ordered sets, but since any well-ordered set can be indexed by ordinals in an order-preserving way, it suffices to
Transfinite_induction
Standard system of axiomatic set theory
"definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership
Zermelo–Fraenkel_set_theory
1940 non-fiction book by H. G. Wells
New World Order is a non-fiction book written by H. G. Wells and published by Secker & Warburg in January 1940. In The New World Order, Wells proposes
The New World Order (Wells book)
The_New_World_Order_(Wells_book)
Type of ordering of a set
In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle
Dense_order
American television series (1999–present)
2026, Law & Order: Special Victims Unit has aired 594 original episodes, well surpassing the episode count of the original Law & Order series. In terms
Law & Order: Special Victims Unit
Law_&_Order:_Special_Victims_Unit
better-quasi-ordering is a well-quasi-ordering. Though well-quasi-ordering is an appealing notion, many important infinitary operations do not preserve well-quasi-orderedness
Better-quasi-ordering
Papal Order of Knighthood of the Holy See
arts. The order has been bestowed upon members of other Christian denominations as well, in addition to those of other religions. This Order was at one
Order_of_St._Sylvester
Smallest ordinal number that, considered as a set, is uncountable
{\displaystyle \Omega } , is the smallest ordinal number that is the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all
First_uncountable_ordinal
incomparabilities) Well-orders, total orders in which every non-empty subset has a least element Well-quasi-orderings, a class of preorders generalizing the well-orders
List of order structures in mathematics
List_of_order_structures_in_mathematics
Algebraic object with an ordered structure
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples
Ordered_field
Measure of how well someone's life is going
temporal order of episodes of well-being matters. Welfare biology, a related field, examines whether all sentient beings are capable of well-being, under
Well-being
Form of mathematical proof
\{(1,n):n\in \mathbb {N} \}} , shown in the picture, is well-ordered by the lexicographic order. Moreover, except for the induction axiom, it satisfies
Mathematical_induction
Axiom of set theory
choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set created by choosing
Axiom_of_choice
Well-quasi-ordering of finite trees
X} is well-quasi-ordered, then the set of rooted trees with labels in X {\displaystyle X} is well-quasi-ordered under the inf-embeddable order defined
Kruskal's_tree_theorem
Construction in order theory
B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order ≤ {\displaystyle \leq } on the Cartesian
Product_order
Concept in order theory
In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least
Join_and_meet
Equivalence of partially ordered sets
In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism
Order_isomorphism
Mathematical ordering of a partial order
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial
Linear_extension
this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little
Specialization_preorder
18th-century Bavarian secret society
influence over public life, and abuses of state power by monarchs. "The order of the day", they wrote in their general statutes, "is to put an end to
Illuminati
On chains and antichains in partial orders
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size
Dilworth's_theorem
Australian national honour
The Order of Australia is an Australian honour which recognises Australian citizens and other persons for outstanding service and achievement. It was
Order_of_Australia
Type of monotone function
other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic. An example
Order_embedding
finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well-ordering if and only if the tree has no infinite branch. It is named
Kleene–Brouwer_order
Partial order in lattice theory
In order theory, the strong set order ≤ s {\displaystyle \leq _{s}} is a partial order over the subsets of a lattice. It is widely used to study monotone
Strong_set_order
Mathematical ranking of a set
In 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
Weak_ordering
Mathematical system
arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers
Second-order_arithmetic
Mathematical result or axiom on order relations
the set of all chains in P {\displaystyle P} . By the well-ordering theorem, we find a well-ordering ⪯ {\displaystyle \preceq } on P {\displaystyle P} .
Hausdorff_maximal_principle
Number representing a continuous quantity
a least element in this ordering. (The standard ordering ≤ {\displaystyle \leq } of the real numbers is not a well-ordering since e.g. an open interval
Real_number
Size of a possibly infinite set
classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and ω 1 {\displaystyle \omega _{1}} is the order type
Cardinal_number
Topology of an ordered vector space
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )}
Order topology (functional analysis)
Order_topology_(functional_analysis)
Proof method in mathematical logic
recursively defined structure, such as formulas, lists, or trees. A well-founded partial order is defined on the structures ("subformula" for formulas, "sublist"
Structural_induction
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous
Prefix_order
Medieval military order
crusading military order for supporting Catholic rule in the Holy Land and the Northern Crusades during the Middle Ages, as well as supplying military
Teutonic_Order
Protestant fraternal order based in Northern Ireland
Orange Institution, commonly known as the Orange Order, is an international Protestant fraternal order based in Northern Ireland and primarily associated
Orange_Order
Particular class of sets which can be described entirely in terms of simpler sets
{\displaystyle L} . It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class V
Constructible_universe
Vector space with a partial order
partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space
Ordered_vector_space
American television series (1990–2010, 2022–present)
by Jury, Law & Order: LA, Law & Order True Crime, and Law & Order: Organized Crime) as well as a television film (Exiled: A Law & Order Movie). The commercial
Law_&_Order
Well drilled to extract crude oil and/or gas
associated petroleum gas along with the oil. A well that is designed to produce only gas may be termed a gas well. Wells are created by drilling down into an oil
Oil_well
consisting of all well-ordered sets of the same order type. To have the same order type is an equivalence relation on the class of well-ordered sets, and
Paradoxes_of_set_theory
British order of chivalry established in 1725
The Most Honourable Order of the Bath is a British order of chivalry founded by King George I on 18 May 1725. Recipients of the Order are usually senior
Order_of_the_Bath
Set whose pairs have minima and maxima
geometric lattices (matroids). These lattice-like structures all admit order-theoretic as well as algebraic descriptions. The sub-field that studies lattices
Lattice_(order)
the order type of a well-ordering W is the set of all well-orderings which are similar to W. The set of ordinal numbers is the set of all order types
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Collection of mathematical objects
choice is equivalent with the fact that a well-order can be defined on every set, where a well-order is a total order such that every nonempty subset has a
Set_(mathematics)
Mathematical function
choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that
Choice_function
Mathematical concept for comparing objects
relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure
Equivalence_relation
Property of a relation on a set
linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that
Connected_relation
of countable unions of scattered orders is a well-quasi-order. The order topology of a scattered order is scattered. The converse implication does not
Scattered_order
Special type of lattice
can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra. Both views and their mutual correspondence
Distributive_lattice
Numerical ordering with a margin of error
In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are
Semiorder
Reversal of the order of elements of a binary relation
operation of complementation as well as with taking suprema and infima. Conversion is also compatible with the ordering of relations by inclusion. If a
Converse_relation
Ordinals in mathematics and set theory
is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an
Large_countable_ordinal
Catholic lay religious order
its nature as a lay religious order, as well as particular terminology evolved from nine centuries of history. The order's membership includes about 13
Sovereign Military Order of Malta
Sovereign_Military_Order_of_Malta
Heritage railway in North Norfolk, England
was authorised by the Wells and Walsingham Light Railway Order 1982, the terms of which were altered under the subsequent Wells and Walsingham Light Railway
Wells and Walsingham Light Railway
Wells_and_Walsingham_Light_Railway
Catholic military order
Order of Knights of the Hospital of Saint John of Jerusalem, commonly known as the Knights Hospitaller (/ˈhɒspɪtələr/), was a Catholic military order
Knights_Hospitaller
Countable ordinal that is the order type of a computable well-ordering of natural numbers
computable well-ordering of natural numbers. An ordinal α {\displaystyle \alpha } is computable if there exists a computable well-ordering ≺ {\displaystyle
Computable_ordinal
Binary relation over a set and itself
endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology
Homogeneous_relation
German logician and mathematician (1871–1953)
developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the
Ernst_Zermelo
Size of subsets in order theory
\delta } that is the order type of a cofinal subset of α . {\displaystyle \alpha .} The cofinality of a set of ordinals or any other well-ordered set is the
Cofinality
Legal/criminal procedural television franchise
Law & Order is a media franchise composed of a number of related American television series created by Dick Wolf and produced by Wolf Entertainment. They
Law_&_Order_(franchise)
Partially ordered vector space, ordered as a lattice
vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first
Riesz_space
English writer (1866–1946)
skeptical to Christianity, HG Wells had surprisingly positive views on Islam. He complimented how Islam brought equity and order to a tribalistic part of the
H._G._Wells
Paradox in set theory
Russell–Myhill paradox The Burali-Forti paradox, about the order type of all well-orderings Curry's paradox (named after Haskell Curry), which does not
Russell's_paradox
Partially ordered set in which all subsets have both a supremum and infimum
lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special class of lattices.
Complete_lattice
Conspiracy theory regarding a totalitarian world government
writer and futurist H. G. Wells went further than progressives in the 1940s, by appropriating and redefining the term "new world order" as a synonym for the
New World Order conspiracy theory
New_World_Order_conspiracy_theory
2025 American horror television series
Anne Rice's Talamasca: The Secret Order, or simply Talamasca: The Secret Order, is an American supernatural horror spy thriller drama television series
Talamasca:_The_Secret_Order
Catholic military order, 1118 to 1312
speculative popular publications surrounding the order's early occupation of the Temple Mount in Jerusalem as well as speculation about what relics the Templars
Knights_Templar
Partially ordered topological space
{\displaystyle X} equipped with a closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle
Partially_ordered_space
Computer markup language
1, 1 1 1)) ) Well-known binary (WKB) representations are typically shown in hexadecimal strings. The first byte indicates the byte order for the data:
Well-known text representation of geometry
Well-known_text_representation_of_geometry
Special subset of a partially ordered set
filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice
Filter_(mathematics)
Type of topology in mathematics
opposite convention also exists.) The following dictionary holds between order-theoretic notions and topological notions: Open sets are upper sets, Closed
Alexandrov_topology
Mathematical result on order relations
In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930, states that every partial
Szpilrajn_extension_theorem
In mathematics, invertible homomorphism
transitive, total, trichotomous, a partial order, total order, well-order, strict weak order, total preorder (weak order), an equivalence relation, or a relation
Isomorphism
Infinite cardinal number
⋯ } {\displaystyle \{1,3,5,7,9,\cdots ;2,4,6,8,10,\cdots \}} is a well-ordering of the set (with cardinality ℵ 0 {\displaystyle \aleph _{0}} ) of positive
Aleph_number
Type of binary relation
R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and
Transitive_relation
Mathematical rule
order. This ordering is a total order: every positive integer appears exactly once somewhere on this list. However, it is not a well-order. In a well-order
Sharkovskii's_theorem
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