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Mathematical ranking of a set
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
Weak_ordering
Mathematical set with an ordering
contained in some total order. Stochastic dominance – Partial order between random variables Strict weak ordering – strict partial order "<" in which the relation
Partially_ordered_set
Order whose elements are all comparable
ordered set then f induces a total ordering on X by setting x1 ≤ x2 if and only if f(x1) ≤ f(x2). The lexicographical order on the Cartesian product of a family
Total_order
Class of mathematical orderings
every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set (or woset). In some
Well-order
Rules that guarantee predictable computer memory operation
therefore no additional safety net is required for weak ordering. In order to maintain weak ordering, write operations prior to a synchronization operation
Consistency_model
Well-quasi-ordering of finite trees
variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first
Kruskal's_tree_theorem
2022 South Korean television series
Weak Hero (Korean: 약한영웅) is a South Korean television series written and directed by Yoo Soo-min with Kim Jin-seok and Park Dan-hee, starring Park Ji-hoon
Weak_Hero
Type of consistency in programming which is based synchronization
to weak ordering. They must label synchronization accesses as acquires or releases, not just as synchronization accesses. Similar to weak ordering, Release
Release_consistency
Number of orderings allowing ties
nonempty subsets. A weak ordering may be obtained from such a partition by choosing one of k ! {\displaystyle k!} total orderings of its subsets. Therefore
Ordered_Bell_number
Partial order on a Coxeter group
right weak Bruhat orderings were studied by Björner (1984). If (W, S) is a Coxeter system with generators S, then the Bruhat order is a partial order on
Bruhat_order
Branch of mathematics
and relations of a partial ordering. These are graph drawings where the vertices are the elements of the poset and the ordering relation is indicated by
Order_theory
processes can observe only one consistent state. The original paper on weak ordering: M. Dubois, C. Scheurich and F. A. Briggs, Memory Access Buffering in
Weak_consistency
Reflexive and transitive binary relation
} is an equivalence; in that case " < {\displaystyle <} " is a strict weak order. The resulting preorder is connected (formerly called total); that is
Preorder
specifically in order theory and functional analysis, an element x {\displaystyle x} of a vector lattice X {\displaystyle X} is called a weak order unit in X
Weak_order_unit
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
Set whose pairs have minima and maxima
the lattice of normal subgroups of a group. The set of first-order terms with the ordering "is more specific than" is a non-modular lattice used in automated
Lattice_(order)
three-way comparison Possible return types: std::weak_ordering, std::strong_ordering and std::partial_ordering to which they all are convertible to. In the
Operators_in_C_and_C++
Topics referred to by the same term
up weak in Wiktionary, the free dictionary. Weak may refer to: "Weak" (AJR song), 2016 "Weak" (Melanie C song), 2011 "Weak" (SWV song), 1993 "Weak" (Skunk
Weak
Software library for the C++ programming language
such comparison operator or comparator function must guarantee strict weak ordering. Apart from these, algorithms are provided for making heap from a range
Standard_Template_Library
Pair of positions in a sequence where two elements are out of sorted order
as lex order by r {\displaystyle r} . The set of permutations on n items can be given the structure of a partial order, called the weak order of permutations
Inversion (discrete mathematics)
Inversion_(discrete_mathematics)
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
Relationship between elements of two sets
transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too
Binary_relation
Mathematical ordering of a partial order
as the ordering principle, OP, and is a weakening of the well-ordering theorem. However, there are models of set theory in which the ordering principle
Linear_extension
Order-preserving mathematical function
f\!\left(y\right)} . To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity
Monotonic_function
Type of binary relation
symmetric preorder Strict weak ordering – a strict partial order in which incomparability is an equivalence relation Total ordering – a connected (total)
Transitive_relation
Glossary of terms used in branch of mathematics
that x approximates y. See also domain theory. Weak order. A partial order ≤ on a set X is a weak order provided that the poset (X, ≤) is isomorphic to
Glossary_of_order_theory
Type of ordering of a set
{\displaystyle \mathbb {Z} [x]} is dense. On the other hand, the linear ordering on the integers is not dense. Georg Cantor proved that every two non-empty
Dense_order
Season of American television series
NYPD's Chief of Detectives which became a recurring part. The episodes "Weak", "Contagious" and "Identity" starred Mary Stuart Masterson as Dr. Rebecca
Law & Order: Special Victims Unit season 6
Law_&_Order:_Special_Victims_Unit_season_6
Partition of vertices of a directed graph
consistent with reachability. This ordering on the weak components can alternatively be interpreted as a weak ordering on the vertices themselves, with
Weak_component
Isomorphism type of ordered sets
standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving
Order_type
Type of monotone function
another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings
Order_embedding
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing
List_of_order_theory_topics
Polyhedron whose vertices represent permutations
strict weak orderings of the set {1 ... n}. So the number of all faces is the n-th ordered Bell number. A face of dimension d corresponds to an ordering with
Permutohedron
characterized as the N-free finite partial orders; they have order dimension at most two. They include weak orders and the reachability relationship in directed
Series-parallel_partial_order
Equivalence of partially ordered sets
other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections. The idea of
Order_isomorphism
Alternative mathematical ordering
circular ordering (Mosher 1996, p. 109). Some authors call such an ordering a total cyclic order (Isli & Cohn 1998, p. 643), a complete cyclic order (Novák
Cyclic_order
Property of elements related by inequalities
the partial order ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobriety criteria are not. Strict weak ordering – Mathematical
Comparability
Mathematical result or axiom on order relations
set of all chains in P {\displaystyle P} . By the well-ordering theorem, we find a well-ordering ⪯ {\displaystyle \preceq } on P {\displaystyle P} . We
Hausdorff_maximal_principle
Index of articles associated with the same name
refer to orderings that describe human preferences for one thing over an other. In mathematics, preferences may be modeled as a weak ordering or a semiorder
Preference_relation
In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array. Every better-quasi-ordering is a well-quasi-ordering
Better-quasi-ordering
Subset of incomparable elements
smallest number of antichains into which the order may be partitioned. An antichain in the inclusion ordering of subsets of an n {\displaystyle n} -element
Antichain
Computing technique
regions) due to the weak ordering. Write-combining does not guarantee that the combination of writes and reads is done in the expected order. For example, a
Write_combining
Order of accesses to computer memory by a CPU
1998) may have weaker 'oostore' memory ordering. RISC-V memory ordering models WMO Weak memory order (default) TSO Total store order (only supported
Memory_ordering
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
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)
Size of subsets in order theory
in R . {\displaystyle \mathbb {R} .} The usual ordering of R {\displaystyle \mathbb {R} } is not order isomorphic to c , {\displaystyle c,} the cardinality
Cofinality
Action of arranging objects into order
Sorting refers to ordering data in an increasing or decreasing manner according to some linear relationship among the data items. ordering: arranging items
Sorting
Consistency model in concurrent computing
memory location must be seen by all processors in the same order. Similar to weak ordering, the release consistency model allows reordering of all memory
Processor_consistency
Index of articles associated with the same name
of directed acyclic graphs Degeneracy ordering of undirected graphs Elimination ordering of chordal graphs Order, the complexity of a structure within
Order_(mathematics)
Mathematical ways to group elements of a set
Partition refinement Point-finite collection Rhyme schemes by set partition Weak ordering (ordered set partition) Knuth, Donald E. (2013), "Two thousand years
Partition_of_a_set
Visual depiction of a partially ordered set
In order theory, a Hasse diagram (/ˈhæsə/; German: [ˈhasə]) is a type of mathematical diagram used to represent a finite partially ordered set, in the
Hasse_diagram
Type of binary relation
well-ordering principle. There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the
Well-founded_relation
Reversal of the order of elements of a binary relation
transitive, connected, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its converse
Converse_relation
Vector space with a partial order
{\displaystyle W.} In this case, the ordering defined by C {\displaystyle C} is called the canonical ordering of L ( X ; W ) . {\displaystyle L(X;W)
Ordered_vector_space
Type of topology in mathematics
these topologies. McCord also showed that these spaces are weak homotopy equivalent to the order complex of the corresponding partially ordered set. Steiner
Alexandrov_topology
Nonempty, upper-bounded, downward-closed subset
I. A weaker notion of order ideal is defined to be a subset of a poset P that satisfies the above conditions 1 and 2. In other words, an order ideal
Ideal_(order_theory)
Existence of certain infima or suprema of a given poset
given by q(x) = (x, x). Naturally, the intended ordering relation for X × X is just the usual product order. q has a lower adjoint q* if and only if all
Completeness_(order_theory)
Special type of lattice
space is a distributive lattice. Young's lattice given by the inclusion ordering of Young diagrams representing integer partitions is a distributive lattice
Distributive_lattice
Property of a relation on a set
different properties. Sources which define both then use pairs of terms such as weakly connected and connected, complete and strongly complete, total and complete
Connected_relation
When a system's behavior depends on timing of uncontrollable events
memory model provides SC for DRF and allows the optimizations of the WO (weak ordering), RCsc (release consistency with sequentially consistent special operations)
Race_condition
Algebraic object with an ordered structure
its standard ordering (which is also its only ordering); the field R {\displaystyle \mathbb {R} } of real numbers with its standard ordering (which is also
Ordered_field
Characterizes the height of any finite partially ordered set
the set into chains. For sets of order dimension two, the two theorems coincide (a chain in the majorization ordering of points in general position in
Mirsky's_theorem
Partially ordered vector space, ordered as a lattice
. {\displaystyle W.} In this case the ordering defined by C {\displaystyle C} is called the canonical ordering of L ( X ; W ) . {\displaystyle \operatorname
Riesz_space
There are equally many countable order types and real numbers
theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals
Cantor–Bernstein_theorem
Mathematical relation inside orderings
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements
Covering_relation
Mathematical property of subsets in order theory
{\displaystyle C} is a cofinal subset of B {\displaystyle B} (with the partial ordering of A {\displaystyle A} applied to B {\displaystyle B} ), then C {\displaystyle
Cofinal_(mathematics)
restrictions Total orders, orderings that specify, for every two distinct elements, which one is less than the other Weak orders, generalizations of total
List of order structures in mathematics
List_of_order_structures_in_mathematics
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
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
In mathematics, invertible homomorphism
special properties, if and only if R is. For example, R is an ordering ≤ and S an ordering ⊑ , {\displaystyle \scriptstyle \sqsubseteq ,} then an isomorphism
Isomorphism
Mathematical proposition equivalent to the axiom of choice
field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel
Zorn's_lemma
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous
Prefix_order
Mathematical ordering with upper bounds
the "reals directed towards x 0 {\displaystyle x_{0}} " but in which the ordering rule only applies to pairs of elements on the same side of x 0 {\displaystyle
Directed_set
Bound lattice in which every element has a complement
is an involution that is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law
Complemented_lattice
1951 book by Kenneth Arrow
one) set of orderings onto a social ordering, a corresponding ordering of the set of social states that applies to each voter. A social ordering of a constitution
Social Choice and Individual Values
Social_Choice_and_Individual_Values
Specially annotated symbol in an object file
defines symbol f and declares it as weak. libbar also defines f and declares it as strong. Depending on the library ordering on the link command line (i.e.
Weak_symbol
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
" Computer 41.7 (2008): 33–38. Adve, Sarita V., and Mark D. Hill. "Weak ordering—a new definition." [1990] Proceedings. The 17th Annual International
Mark_D._Hill
A type of item in a relational database, in computing
In a relational database, a weak entity is an entity that cannot be uniquely identified by its attributes alone; therefore, it must use a foreign key in
Weak_entity
Natural number
digits: 7, 5, 12, 17, 29, 46, 75... 75 is the count of the number of weak orderings on a set of four items. Excluding the infinite sets, there are 75 uniform
75_(number)
Ideals in a Boolean algebra can be extended to prime ideals
subset ordering, the "maximal filter theorem" is called the ultrafilter lemma. Summing up, for Boolean algebras, the weak and strong MIT, the weak and strong
Boolean_prime_ideal_theorem
Method in Itô calculus
{\displaystyle g({\hat {X}}_{N})-g(X_{T})} . Strong order γ s {\displaystyle \gamma _{s}} convergence implies weak order γ w ≥ γ s {\displaystyle \gamma _{w}\geq
Euler–Maruyama_method
Theorem in order theory
Dushnik–Miller theorem is a result in order theory stating that every countably infinite linear order has a non-identity order embedding into itself. It is named
Dushnik–Miller_theorem
Partially ordered set in which all subsets have both a supremum and infimum
fact much more specific. For this reason, it can be useful to consider weaker notions of morphisms, such as those that are only required to preserve all
Complete_lattice
Partially ordered set equipped with a rank function
ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering
Graded_poset
Algebraic structure used in logic
Heyting algebras for any n (but they are not MV-algebras for n ≥ 5). The ordering ≤ {\displaystyle \leq } on a Heyting algebra H can be recovered from the
Heyting_algebra
Mathematical result on order relations
extending its empty partial order, finding a cofinal well-order, and choosing the minimum element from that well-ordering. Arrow stated that every preorder
Szpilrajn_extension_theorem
time scales with the time step δ {\displaystyle \delta } . It has also weak order 1, meaning that the error on the statistics of the solution scales with
Runge–Kutta_method_(SDE)
Banach space with a compatible structure of a lattice
in functional analysis and order theory, a Banach lattice (X,‖·‖) is a complete normed vector space with a lattice order, ≤ {\displaystyle \leq } , such
Banach_lattice
Length of time which a note can last
poetry: iamb (weak–strong), anapest (weak–weak–strong), trochee (strong–weak), dactyl (strong–weak–weak), and amphibrach (weak–strong–weak), which may overlap
Duration_(music)
In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff
Weak_Hausdorff_space
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
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
Mathematical property excluding equality
and not equal to"). More generally, a strict partial order, strict total order, and strict weak order exclude equality and equivalence. When comparing numbers
Strict
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)
analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial order ≤ {\displaystyle
Topological_vector_lattice
Subset of a preorder that contains all larger elements
{\downarrow \!y}.} Thus, the above construction can be used to replace a given ordering by set inclusion and also yields advantages such as that a least upper
Upper_and_lower_sets
Relationship between items in a set
In mathematics, this is known as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects
Ranking
Special subset of a partially ordered set
passing from a preordering to associated partial ordering. Historically, filters generalized to order-theoretic lattices before arbitrary partial orders
Filter_(mathematics)
Function for sorting in C++ standard library
comparison predicate. This comparison predicate must define a strict weak ordering on the elements of the sequence to be sorted. The third argument is
Sort_(C++)
WEAK ORDERING
WEAK ORDERING
Boy/Male
Hindu
Peak
Girl/Female
Indian
Peak
Boy/Male
Spanish
Weak.
Boy/Male
Arabic, Muslim
Weak
Girl/Female
Tamil
Peak
Boy/Male
Arabic
One who has Weak Eyes
Boy/Male
Arabic, Muslim
Weak
Boy/Male
Tamil
Peak
Girl/Female
Hindu, Indian, Traditional
Peak
Biblical
weak; slacked
Boy/Male
Biblical
Weak, slacked.
Boy/Male
Hindu, Indian
Week
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Peak
Boy/Male
Australian, Hindu, Indian
Peak
Surname or Lastname
English
English : variant of Week.
Surname or Lastname
English (Northumbria)
English (Northumbria) : topographic name for someone who lived by the Wear river in northern England. The river name is ancient, occuring in the form Vedra in Ptolemy’s Geographia; it is probably a Celtic word meaning ‘water’.English (Northumbria) : topographic name for someone who lived near a dam or weir, a variant spelling of Ware 1, or a habitational name from a place called Weare, in Devon and Somerset, from Old English wær, wer ‘weir’.
Surname or Lastname
English
English : variant spelling of Leake.
Surname or Lastname
English
English : topographic name for someone living by a pointed hill (or regional name from the Peak District (Old English Pēaclond) in Derbyshire), named with Old English pēac ‘peak’, ‘pointed hill’ (found only in place names). This word is not directly related to Old English pīc ‘point’, ‘pointed hill’, which yielded Pike; there is, however, some evidence of confusion between the two surnames.Possibly also Irish : reduced form of McPeak.Major concentrations of the surname Peak are found in Staffordshire and the West Country of England. Among the earliest known bearers are Richard del Pech or del Pek (d. 1196), son of Rannulf, sheriff of Nottingham, and Willielmus Piec (Winchester 1194). A century later, c.1284, a certain Richard del Peke settled in Denbighshire (now part of Clwyd), Wales, receiving lands from Henry de Lacey, earl of Lincoln, in return for helping to control the region. His descendants, who bear the name Peak(e), can be traced to the present day, and are found in New Zealand and Canada as well as in Britain. Peake is also the name of a family descended from John Pyke, who paid rent to the abbot of Leicester in 1477. The name took various forms, such as Peke and Pick, eventually becoming established as Peak in the 17th century.
Surname or Lastname
English
English : variant of Wick, specifically a habitational name from any of various places called Week or Weeke, notably in Cornwall, Hampshire, and Somerset.Americanized spelling of Norwegian or Swedish Vik.
Girl/Female
Australian, Christian, French, German, Hebrew
Hair; Lovelorn; Delicate; Weak
WEAK ORDERING
WEAK ORDERING
Boy/Male
Muslim
Agreement, Covenant
Girl/Female
Latin American Swedish
Sign.
Boy/Male
Greek Latin
God of medicine.
Girl/Female
Tamil
Related with season
Boy/Male
Indian
Charming and beautiful
Girl/Female
Tamil
Pournami | பௌரமாஂமீ
Day of the full Moon
Girl/Female
Arabic
Goodness.
Girl/Female
Tamil
Path between trees
Boy/Male
Hindu
Kindness, Beneficence, Highest level of Iman
Girl/Female
Latin
Understanding.
WEAK ORDERING
WEAK ORDERING
WEAK ORDERING
WEAK ORDERING
WEAK ORDERING
v. i.
Lacking in elements of political strength; not wielding or having authority or energy; deficient in the resources that are essential to a ruler or nation; as, a weak monarch; a weak government or state.
v. i.
Not able to withstand temptation, urgency, persuasion, etc.; easily impressed, moved, or overcome; accessible; vulnerable; as, weak resolutions; weak virtue.
a.
To make or become weak; to weaken.
v. t.
To cause or make by friction or wasting; as, to wear a channel; to wear a hole.
v. i.
Not able to sustain a great weight, pressure, or strain; as, a weak timber; a weak rope.
v. i.
Not having power to convince; not supported by force of reason or truth; unsustained; as, a weak argument or case.
v. i.
Feeble of mind; wanting discernment; lacking vigor; spiritless; as, a weak king or magistrate.
v. i.
Not firmly united or adhesive; easily broken or separated into pieces; not compact; as, a weak ship.
v. i.
Lacking ability for an appropriate function or office; as, weak eyes; a weak stomach; a weak magistrate; a weak regiment, or army.
a.
Having weak knees; hence, easily yielding; wanting resolution.
v. i.
Tending towards lower prices; as, a weak market.
v. i.
Wanting in power to influence or bind; as, weak ties; a weak sense of honor of duty.
v. i.
To rise or extend into a peak or point; to form, or appear as, a peak.
a.
Having a weak mind, either naturally or by reason of disease; feebleminded; foolish; idiotic.
n.
The upper aftermost corner of a fore-and-aft sail; -- used in many combinations; as, peak-halyards, peak-brails, etc.
v. i.
Not stiff; pliant; frail; soft; as, the weak stalk of a plant.
v. i.
Wanting in point or vigor of expression; as, a weak sentence; a weak style.
v. i.
Not able to resist external force or onset; easily subdued or overcome; as, a weak barrier; as, a weak fortress.
v.
A crack, crevice, fissure, or hole which admits water or other fluid, or lets it escape; as, a leak in a roof; a leak in a boat; a leak in a gas pipe.
v. i.
Not thoroughly or abundantly impregnated with the usual or required ingredients, or with stimulating and nourishing substances; of less than the usual strength; as, weak tea, broth, or liquor; a weak decoction or solution; a weak dose of medicine.