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Group with a compatible partial order
space Ordered vector space – Vector space with a partial order Partially ordered ring – Ring with a compatible partial order Partially ordered space –
Partially_ordered_group
Mathematical set with an ordering
graph ordered by reachability. The set of subspaces of a vector space ordered by inclusion. For a partially ordered set P, the sequence space containing
Partially_ordered_set
Partially ordered topological space
In mathematics, a partially ordered space (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle
Partially_ordered_space
Ring with a compatible partial order
targets Ordered topological vector space Ordered vector space – Vector space with a partial order Partially ordered space – Partially ordered topological
Partially_ordered_ring
Algebraic object with an ordered structure
space Ordered vector space – Vector space with a partial order Partially ordered ring – Ring with a compatible partial order Partially ordered space –
Ordered_field
Vector space with a partial order
ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations
Ordered_vector_space
compatible partial order Partially ordered space – Partially ordered topological space Riesz space – Partially ordered vector space, ordered as a lattice, also
Ordered_ring
targets Ordered ring Ordered vector space – Vector space with a partial order Partially ordered space – Partially ordered topological space Riesz space – Partially
Ordered topological vector space
Ordered_topological_vector_space
Certain topology in mathematics
continuum Order topology (functional analysis) Partially ordered space Lynn, I. L. (1962). "Linearly orderable spaces". Proceedings of the American Mathematical
Order_topology
Partially ordered vector space, ordered as a lattice
Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are
Riesz_space
Type of random variable ordering
P α ) α ∈ F {\displaystyle ({P}_{\alpha })_{\alpha \in F}} on partially ordered space ( E , ⪯ ) {\displaystyle (E,\preceq )} indexed with α ∈ F {\displaystyle
Stochastic_ordering
Mathematical proposition equivalent to the axiom of choice
set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains
Zorn's_lemma
Order whose elements are all comparable
sets. Applied to the vector space Rn, each of these make it an ordered vector space. See also examples of partially ordered sets. A real function of n
Total_order
Subset of incomparable elements
a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a finite partially ordered
Antichain
Generalized alphabetical order
Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. The words in
Lexicographic_order
Alternative mathematical ordering
generalization to a locally partially ordered space is studied in Roll (1993); see also Directed topology. A cyclically ordered group is a set with both
Cyclic_order
Class of mathematical orderings
that is also maximum of the whole set. A well-ordered set as topological space is a first-countable space if and only if it has order type less than or
Well-order
Order-preserving mathematical function
graph is a maximal monotone set. Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers. The
Monotonic_function
Definition of continuity for functions between posets
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott)
Scott_continuity
Branch of mathematics
partially ordered set is attributed to Garrett Birkhoff in the second edition of his influential book Lattice Theory. This section introduces ordered
Order_theory
Set whose pairs have minima and maxima
subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called
Lattice_(order)
Special subset of a partially ordered set
In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear
Filter_(mathematics)
On chains and antichains in partial orders
and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals
Dilworth's_theorem
Banach space with a compatible structure of a lattice
abstract (L)-spaces. Banach space – Normed vector space that is complete Normed vector lattice Riesz space – Partially ordered vector space, ordered as a lattice
Banach_lattice
Visual depiction of a partially ordered set
represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle
Hasse_diagram
Partially reusable launch system and space plane
The Space Shuttle is a retired, partially reusable low Earth orbital spacecraft system operated from 1981 to 2011 by the U.S. National Aeronautics and
Space_Shuttle
Set of vectors used to define coordinates
The set R 2 {\displaystyle \mathbb {R} ^{2}} of the ordered pairs of real numbers is a vector space under the operations of component-wise addition ( a
Basis_(linear_algebra)
Well-quasi-ordering of finite trees
states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. A finitary application
Kruskal's_tree_theorem
In mathematics, a locally finite poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements
Locally_finite_poset
Visualisation method for hierchical data
advantage of treemaps is that, by construction, they make efficient use of space. As a result, they can legibly display thousands of items on the screen
Treemapping
Mathematical ranking of a set
generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are
Weak_ordering
Type of topology in mathematics
observed an equivalence between partially ordered sets and spaces that were precisely the T0 versions of the spaces that Alexandrov had introduced. P
Alexandrov_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)
Theorem in economics
contour sets of A and B are identical). He proved that, for every partially-ordered space ( X , ≻ ) {\displaystyle (X,\succ )} that is perfectly-separable
Utility representation theorem
Utility_representation_theorem
Topics referred to by the same term
refer to: Positive cone of an ordered field Positive cone of an ordered vector space Positive cone of a partially ordered group This disambiguation page
Positive_cone
Simplicial set constructed from the objects and morphisms of a small category
where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P)
Nerve_(category_theory)
Property of elements related by inequalities
x{\cancel {\overset {<}{\underset {>}{=}}}}y} is true. A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn
Comparability
Concept in order theory
specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S
Join_and_meet
Size of subsets in order theory
mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of
Cofinality
Type of ordering of a set
covering relation is empty. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real
Dense_order
Mathematical property of subsets in order theory
partially ordered set A {\displaystyle A} admits a totally ordered cofinal subset, then we can find a subset B {\displaystyle B} that is well-ordered
Cofinal_(mathematics)
Mathematical operator
operators on partially ordered sets.) The topological closure of a subset X of a topological space consists of all points y of the space, such that every
Closure_operator
Method of construction of the real numbers
mathematician Norberto Cuesta Dutari [es]. More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding
Dedekind_cut
Mathematical phrase
used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete
Complete_partial_order
Equivalence of partially ordered sets
monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can
Order_isomorphism
Subset of a preorder that contains all larger elements
In mathematics, an upper set S {\displaystyle S} of a partially ordered set X {\displaystyle X} is a subset such that if s is in S and if x in X is larger
Upper_and_lower_sets
1968 film by Stanley Kubrick
featuring extraterrestrials List of films featuring space stations List of incomplete or partially lost films Voyage to the End of the Universe Jason Sperb's
2001:_A_Space_Odyssey
Construction in order theory
on the Cartesian product of totally ordered sets Ordinal sum of partial orders Ordered vector space – Vector space with a partial order Neggers, J.; Kim
Product_order
1986 breakup of American orbiter
the second disaster. The Space Shuttle was a partially reusable spacecraft operated by the US National Aeronautics and Space Administration (NASA). It
Space Shuttle Challenger disaster
Space_Shuttle_Challenger_disaster
2020s class of partially reusable spacecraft
of partially reusable spacecraft developed, manufactured, and operated by the American space company SpaceX for flights to the International Space Station
SpaceX_Dragon_2
German mathematician
Tammer is the author of books including: Variational Methods in Partially Ordered Spaces (with Alfred Göpfert, Hassan Riahi, and Constantin Zălinescu, Springer
Christiane_Tammer
Term in the mathematical area of order theory
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
Duality_(order_theory)
Characterizes the height of any finite partially ordered set
combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of
Mirsky's_theorem
Mathematical ordering with upper bounds
{\displaystyle x_{0}} ; however, it still wouldn't be partially ordered. This example can be generalized to a metric space ( X , d ) {\displaystyle (X,d)} by defining
Directed_set
Partial order with well-ordered predecessors
In set theory, a tree is a partially ordered set ( T , < ) {\displaystyle (T,<)} such that for each t ∈ T {\displaystyle t\in T} , the set { s ∈ T : s
Tree_(set_theory)
Video game series
The ending of Ignition directly sets up the opening of Dead Space 2, where Delille is ordered to find and free Isaac Clarke from an EarthGov asylum, only
Dead_Space
Swedish American mathematician
Bochner with a dissertation entitled "Measure and Integral in Partially Ordered Spaces". During this time, Lagerstrom was also a mathematics instructor
Paco_Lagerstrom
Special type of lattice
and z in L: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite
Distributive_lattice
Property of a partially ordered set
is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset
Least-upper-bound_property
Algebraic structure used in logic
terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω. The open sets of any topological space form a complete
Heyting_algebra
Partially ordered set in which all subsets have both a supremum and infimum
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally
Complete_lattice
British mathematician
June 2014. Stralka, Albert (December 1980). "A partially ordered space which is not a Priestley space". Semigroup Forum. 20 (1). Springer: 293–297. doi:10
Hilary_Priestley
Group with translationally invariant total order
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant
Linearly_ordered_group
Mathematical relation inside orderings
mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements
Covering_relation
Reflexive and transitive binary relation
q {\displaystyle p\wedge q} , and so is q {\displaystyle q} . The partially ordered set ( X / ⇔ , ⇐ ) {\displaystyle \left(X/\Leftrightarrow ,\Leftarrow
Preorder
Theorems generalizing the Brouwer fixed-point theorem
(2019) presented some interesting results on the stability of partially ordered metric spaces for coupled fixed point iteration procedures for mixed monotone
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
Relationship between elements of two sets
the codomain. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x,y), where x is an element of X and y is an element of Y. It encodes
Binary_relation
Void between celestial bodies
The cost of access to space has declined since 2013. Partially reusable rockets such as the Falcon 9 have lowered access to space below $3,500 per kg.
Outer_space
Condition in order theory and topology
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable
Countable_chain_condition
Existence of certain infima or suprema of a given poset
properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the
Completeness_(order_theory)
model: its 'geometric semantics'. Any such model admits a natural partially ordered space (or pospace) structure i.e. a topology and a partial order. The
Directed_algebraic_topology
where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets
Specialization_preorder
Mathematical definition
locally convex topological vector lattice. Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect
Regularly_ordered
Collection of mathematical objects of finite size
any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size". A subset S of a partially ordered
Bounded_set
Isomorphism type of ordered sets
In mathematics, especially in set theory and order theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic
Order_type
Mathematical ordering of a partial order
linear extension may be viewed as an order-preserving bijection from a partially ordered set P {\displaystyle P} to a chain C {\displaystyle C} on the same
Linear_extension
British science fiction TV series (1975–1977)
1975. Due to the production delays, CBS cancelled its order for Space: 1999 and ordered Planet of the Apes for the 1974–1975 season instead, and neither
Space:_1999
Mathematical result or axiom on order relations
1914. It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with
Hausdorff_maximal_principle
order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple
Series-parallel_partial_order
Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty. In descriptive set theory
Inductive_set
Graph linking pairs of comparable elements in a partial order
are not comparable to each other in a partial order. For any strict partially ordered set (S,<), the comparability graph of (S, <) is the graph (S, ⊥) of
Comparability_graph
Topics referred to by the same term
(mathematics), a range of numbers Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets A statistical level of
Interval
inequality, extreme value and mathematical optimization. Partially ordered set Preorder Totally ordered set Total preorder Chain Trichotomy Extended real number
List_of_order_theory_topics
Greatest lower bound and least upper bound
(abbreviated inf; pl.: infima) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the greatest element in P {\displaystyle
Infimum_and_supremum
Particular correspondence between two partially ordered sets
connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical
Galois_connection
Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
the category of Heyting algebras and the category of Esakia spaces. For a partially ordered set (X, ≤) and for x∈ X, let ↓x = {y∈ X : y≤ x} and let ↑x
Esakia_space
Smallest transitive relation containing a given binary relation
relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically
Transitive_closure
1968 science fiction novel by Arthur C. Clarke
2001: A Space Odyssey is a 1968 science fiction novel by English writer Arthur C. Clarke. It was developed concurrently with Stanley Kubrick's film version
2001:_A_Space_Odyssey_(novel)
Natural number
number. Mertens function (219) = 4, a record high. There are 219 partially ordered sets on four labeled elements. 219 is the smallest number that can
219_(number)
Smallest complete lattice containing a partial order
specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named
Dedekind–MacNeille_completion
States of matter for water as a solid
are the plausible space group candidates. Both Yamane et al.'s and Gasser et al.'s results suggested a partially hydrogen-ordered structure. Gasser et
Phases_of_ice
Topics referred to by the same term
number theory Ordinate in mathematics, the y element of an ordered pair (x, y) Partially ordered set Complete partial order Permutation, the act of arranging
Order
Lattice formed by all integer partitions
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers
Young's_lattice
Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets AM-space – Concept in order
Abstract_L-space
Extreme element of a preorder
{\displaystyle S} is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum
Maximal_and_minimal_elements
Partially ordered set equipped with a rank function
mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N
Graded_poset
Mathematical concept for comparing objects
well-founded and has no infinite antichains. Let X {\displaystyle X} be well partially ordered. A (necessarily finite) sequence ( x 1 , x 2 , … , x n ) {\displaystyle
Well-quasi-ordering
Topological vector lattice
Concept in order theory Normed lattice Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect
Fréchet_lattice
Type of monotone function
special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute
Order_embedding
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PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
PARTIALLY ORDERED-SPACE
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