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mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle
Reflexive_closure
Operation on the subsets of a set
intersection of reflexive relations is reflexive, we define the reflexive closure of R {\displaystyle R} on A {\displaystyle A} as the smallest reflexive relation
Closure_(mathematics)
Binary relation that relates every element to itself
set X {\displaystyle X} is reflexive if it relates every element of X {\displaystyle X} to itself. An example of a reflexive relation is the relation "is
Reflexive_relation
Mathematical set with an ordering
corresponding non-strict partial order ≤ {\displaystyle \leq } is the reflexive closure given by: a ≤ b if a < b or a = b . {\displaystyle a\leq b{\text{
Partially_ordered_set
Smallest transitive relation containing a given binary relation
transitive closure on distributed systems based on the MapReduce paradigm. Ancestral relation Deductive closure Reflexive closure Symmetric closure Transitive
Transitive_closure
Reflexive and transitive binary relation
in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest that preorders are
Preorder
R^{\operatorname {T} }.} Transitive closure – Smallest transitive relation containing a given binary relation Reflexive closure Franz Baader and Tobias Nipkow
Symmetric_closure
Order whose elements are all comparable
complement of the converse of ≤ {\displaystyle \leq } ). Conversely, the reflexive closure of a strict total order < {\displaystyle <} is a (non-strict) total
Total_order
Mathematical ranking of a set
associated reflexive relation is its reflexive closure, a (non-strict) partial order ≤ . {\displaystyle \,\leq .} The two associated reflexive relations
Weak_ordering
Well-quasi-ordering of finite trees
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Kruskal's_tree_theorem
Binary relation over a set and itself
containing R. Reflexive transitive closure, R* Defined as R* = (R+)=, the smallest preorder containing R. Reflexive transitive symmetric closure, R≡ Defined
Homogeneous_relation
Relationship between elements of two sets
set X {\displaystyle X} may be subjected to closure operations like: Reflexive closure the smallest reflexive relation over X {\displaystyle X} containing
Binary_relation
Type of ordering of a set
Sufficient conditions for a binary relation R on a set X to be dense are: R is reflexive; R is coreflexive; R is quasireflexive; R is left or right Euclidean;
Dense_order
Subset of a preorder that contains all larger elements
preordered set ( X , ≤ ) , {\displaystyle (X,\leq ),} the upper closure or upward closure of x {\displaystyle x} is defined by ↑ x = { u ∈ X : x ≤ u } {\displaystyle
Upper_and_lower_sets
Visual depiction of a partially ordered set
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Hasse_diagram
Type of binary relation
∈). A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has
Well-founded_relation
Special type of lattice
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Distributive_lattice
Construction in order theory
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Product_order
Isomorphism type of ordered sets
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Order_type
Property of elements related by inequalities
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Comparability
Special subset of a partially ordered set
of terminology. For such posets, downward direction and upward closure reduce to: Closure under finite intersections If A, B ∈ F, then so too is A ∩ B ∈
Filter_(mathematics)
Generalised alphabetical order
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Lexicographic_order
Type of topology in mathematics
Interior and closure algebraic characterizations: The interior operator distributes over arbitrary intersections of subsets. The closure operator distributes
Alexandrov_topology
Size of subsets in order theory
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Cofinality
Set theory concept
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Club_set
Concept in order theory
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Join_and_meet
Type of logical relation
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Total_relation
Equivalence of partially ordered sets
respectively, to the three defining characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomorphism is an equivalence
Order_isomorphism
Mathematical result on order relations
R} is often abbreviated as x R y . {\displaystyle xRy.} A relation is reflexive if x R x {\displaystyle xRx} holds for every element x ∈ X ; {\displaystyle
Szpilrajn_extension_theorem
On chains and antichains in partial orders
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Dilworth's_theorem
Subset of incomparable elements
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Antichain
Category of formal programming language semantics
function, and (2) a Curry-Feys standardization lemma for the transitive-reflexive closure of the single-step relation, which replaces the non-deterministic
Operational_semantics
Order-preserving mathematical function
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Monotonic_function
Characterizes the height of any finite partially ordered set
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Mirsky's_theorem
Type of monotone function
For example: (Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric and transitive) binary relation. An order embedding A →
Order_embedding
suggested by the name, the specialization preorder is a preorder, i.e. it is reflexive and transitive. The equivalence relation determined by the specialization
Specialization_preorder
Partially ordered set equipped with a rank function
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Graded_poset
Mathematical ordering of a partial order
sets is a linear extension of their product order. A partial order is a reflexive, transitive and antisymmetric relation. Given any partial orders ≤ {\displaystyle
Linear_extension
Certain topology in mathematics
general (there are open sets, for example the even numbers from ω, whose closure is not open). The topological spaces ω1 and its successor ω1+1 are frequently
Order_topology
Partially ordered set in which all subsets have both a supremum and infimum
connection from the relation, which then leads to two dually isomorphic closure systems. Closure systems are intersection-closed families of sets. When ordered
Complete_lattice
Nonempty, upper-bounded, downward-closed subset
m ∨ a is not in F. One can construct an ideal N by taking the downward closure of the set of all binary joins of this form, i.e. N = { x | x ≤ m ∨ a for
Ideal_(order_theory)
Mathematical ordering with upper bounds
{\displaystyle A} is inherited from P {\displaystyle P} ; for this reason, reflexivity and transitivity need not be required explicitly. A directed subset of
Directed_set
Branch of mathematics
partial order if it is reflexive, antisymmetric, and transitive, that is, if for all a, b and c in P, we have that: a ≤ a (reflexivity) if a ≤ b and b ≤ a
Order_theory
Mathematical operation
{T}}x} so that R ; R T {\displaystyle R\mathbin {;} R^{\textsf {T}}} is a reflexive relation or I ⊆ R ; R T {\displaystyle \mathrm {I} \subseteq R\mathbin
Composition_of_relations
Mathematical property of subsets in order theory
topology. The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if B {\displaystyle
Cofinal_(mathematics)
Bound lattice in which every element has a complement
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Complemented_lattice
Reversal of the order of elements of a binary relation
compatible with the ordering of relations by inclusion. If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, connected
Converse_relation
Mathematical result or axiom on order relations
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Hausdorff_maximal_principle
Alternative mathematical ordering
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Cyclic_order
Class of mathematical orderings
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Well-order
Mathematical relation inside orderings
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Covering_relation
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Set whose pairs have minima and maxima
Pointless topology Lattice of subgroups Spectral space Invariant subspace Closure operator Abstract interpretation Subsumption lattice Fuzzy set theory Algebraizations
Lattice_(order)
There are equally many countable order types and real numbers
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Cantor–Bernstein_theorem
British philosopher and filmmaker
another but rather in different relationships to openness and closure. In Reflexivity, Lawson argued that self-reference was central to contemporary
Hilary_Lawson
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Better-quasi-ordering
Lattice formed by all integer partitions
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Young's_lattice
Mathematical concept for comparing objects
well-quasi-ordering on a set X {\displaystyle X} is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinite sequence of elements
Well-quasi-ordering
odd). Then the poset of cells of L, ordered by the inclusion of their closures, is Eulerian. Let W be a Coxeter group with Bruhat order. Then (W,≤) is
Eulerian_poset
Algebraic object with an ordered structure
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Ordered_field
Mathematical operator
In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal
Closure_operator
Property of a relation on a set
A relation is strongly connected if, and only if, it is connected and reflexive. A connected relation on a set X {\displaystyle X} cannot be antitransitive
Connected_relation
First compartment of ruminant stomach
into the rumen stomach compartment, as it is instead bypassed by the reflexive closure of the esophageal groove. The most abundant bacteria present in the
Rumen
Construction in order theory
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Star_product
Existence of certain infima or suprema of a given poset
lower adjoint with the function that maps any subset of X to its lower closure (again an adjunction for the inclusion of lower sets in the powerset),
Completeness_(order_theory)
Partial order with well-ordered predecessors
\omega _{1}} . If ( T , < ) {\displaystyle (T,<)} is a tree, then the reflexive closure ≤ {\displaystyle \leq } of < {\displaystyle <} is a prefix order on
Tree_(set_theory)
Graph linking pairs of comparable elements in a partial order
partially ordered set, take the directed acyclic graph, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph
Comparability_graph
Generalization of the concept of subsequence to the case of nets
In other words, every cluster point of a net in a subset belongs to the closure of that set. If x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in
Subnet_(mathematics)
Algebraic structure used in logic
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Heyting_algebra
antisymmetric, transitive, reflexive, and downward total, i.e., for all a, b, and c in P, we have that: a ≤ a (reflexivity); if a ≤ b and b ≤ a then a
Prefix_order
Mathematical proposition equivalent to the axiom of choice
Partially ordered set A set P equipped with a binary relation ≤ that is reflexive (x ≤ x for every x), antisymmetric (if both x ≤ y and y ≤ x hold, then
Zorn's_lemma
if so compute its transitive closure, in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the
Series-parallel_partial_order
Ideals in a Boolean algebra can be extended to prime ideals
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Boolean_prime_ideal_theorem
order of functions Galois connection Order embedding Order isomorphism Closure operator Functions that preserve suprema/infima Dedekind completion Ideal
List_of_order_theory_topics
Mathematical operation
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Demonic_composition
Type of binary relation
not be reflexive. When it is, it is called a preorder. For example, on set X = {1,2,3}: R = { (1,1), (2,2), (3,3), (1,3), (3,2) } is reflexive, but not
Transitive_relation
Term in the mathematical area of order theory
lower bounds (infima, ∧) Upper sets and lower sets Ideals and filters Closure operators and kernel operators. Examples of notions which are self-dual
Duality_(order_theory)
Banach space with a compatible structure of a lattice
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Banach_lattice
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Locally_finite_poset
Partially ordered topological space
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Partially_ordered_space
Glossary of terms used in branch of mathematics
elements of P that are comparable under ≤ (and, in particular, under its reflexive reduction <). Complete Boolean algebra. A Boolean algebra that is a complete
Glossary_of_order_theory
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Numerical ordering with a margin of error
in this way meets the three requirements of a partial order that it be reflexive, antisymmetric, and transitive. Conversely, suppose that ( X , ≤ ) {\displaystyle
Semiorder
Vector space with a partial order
( a = c and b = d ) {\displaystyle (a=c{\text{ and }}b=d)} (the reflexive closure of the direct product of two copies of R {\displaystyle \mathbb {R}
Ordered_vector_space
Algebraic structure modeling logical operations
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Boolean_algebra_(structure)
Reflex at the back of the throat
back of the throat. It, along with other aerodigestive reflexes such as reflexive pharyngeal swallowing, prevents objects in the oral cavity from entering
Pharyngeal_reflex
simplification ordering. The converse, the symmetric closure, the reflexive closure, and the transitive closure of a rewrite relation is again a rewrite relation
Rewrite_order
Total order in computer science
path ordering (>) can be defined as follows: where (≥) denotes the reflexive closure of the mpo (>), { s1,...,sm } denotes the multiset of s’s subterms
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
p\in {\mathcal {P}}} . If the topology on X is locally convex then the closure of a normal cone is a normal cone. If C is a normal cone in X and B is
Ordered topological vector space
Ordered_topological_vector_space
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
List of order structures in mathematics
List_of_order_structures_in_mathematics
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Topological_vector_lattice
Relation specifying a rewrite for each object, compatible with a reduction relation
{\displaystyle {\overset {+}{\to }}} is the transitive closure of → {\displaystyle \to } (but not the reflexive closure). In addition the normal forms of the strategy
Reduction_strategy
{\displaystyle \left(X,\tau _{\operatorname {O} }\right)} is reflexive. Every reflexive locally convex vector lattice is order complete and a complete
Locally_convex_vector_lattice
Partially ordered vector space, ordered as a lattice
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Riesz_space
Topology of an ordered vector space
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Order topology (functional analysis)
Order_topology_(functional_analysis)
Uniqueness of countable dense linear orders
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Cantor's_isomorphism_theorem
Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov
Normed_vector_lattice
Australian animated web series
Caine returns from the void and makes peace with the group, providing closure to them via a presentation of their human selves' lives taken from their
The_Amazing_Digital_Circus
Set theory concept
preorder ≤ {\displaystyle \leq } on X {\displaystyle X} (a transitive and reflexive relation on X {\displaystyle X} ) that is strongly connected (meaning
Prewellordering
Relationship between two sets, defined by a set of ordered pairs
irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric. "is sister of" is neither reflexive (e.g. Pierre Curie is not a sister of himself)
Relation_(mathematics)
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
Boy/Male
Indian, Punjabi, Sikh
One who is Aware and Reflective
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
Female
Scandinavian
Feminine form of Scandinavian Vilhelm, VILHELMA means "will-helmet."
Boy/Male
Spanish
Born at Christmas.
Boy/Male
Arabic, Muslim
Power of Allah
Girl/Female
British, English, Gujarati, Hindu, Indian, Romanian
Lot of Love
Biblical
drops of the sea
Boy/Male
American, British, English
Guardian of the Deer
Boy/Male
Indian, Sanskrit
Slayer of Canda and Munda
Male
Basque
, Ascension.
Boy/Male
Anglo, British, English, German
Noble
Boy/Male
British, English, Greek
Wealthy Defender; Gift of God
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
REFLEXIVE CLOSURE
a.
Throwing back images; as, a reflective mirror.
a.
Repletive.
v. t.
To accustom; -- used reflexively.
v. t. & i.
To eat to excess; -- often with a reflexive.
a.
Implying censure.
n.
The state or condition of being reflected.
a.
Capable of exercising thought or judgment; as, reflective reason.
a.
Having for its direct object a pronoun which refers to the agent or subject as its antecedent; -- said of certain verbs; as, the witness perjured himself; I bethought myself. Applied also to pronouns of this class; reciprocal; reflective.
a.
Reflexive; reciprocal.
a.
Addicted to introspective or meditative habits; as, a reflective person.
a.
Inflective.
a.
Not reflective.
a.
Tending to make replete; filling.
a.
Inflexible.
n.
See Reflection.
v. t.
To carry; to conduct; -- with a reflexive pronoun.
n.
To behave; -- with the reflexive; as, he conducted himself well.
pron.
Themselves; -- used reflexively.
v. t.
To betake; to remove; -- in a reflexive use.
a.
Bending or turned backward; reflective; having respect to something past.