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Smallest transitive relation containing a given binary relation
mathematics, the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite
Transitive_closure
Directed graph with no directed cycles
also contains a longer directed path from u to v. Like the transitive closure, the transitive reduction is uniquely defined for DAGs. In contrast, for a
Directed_acyclic_graph
Operation on the subsets of a set
{\displaystyle (y,z)} to ( x , z ) {\displaystyle (x,z)} , we define the transitive closure of R {\displaystyle R} on A {\displaystyle A} as the smallest relation
Closure_(mathematics)
Class of mathematical set whose elements are all subsets
{\mathcal {P}}(X)} . The power set of a transitive set without urelements is transitive. The transitive closure of a set X {\displaystyle X} , denoted
Transitive_set
Type of binary relation
the transitive extension of Ri would be Ri + 1. The transitive closure of R, denoted by R* or R∞ is the set union of R, R1, R2, ... . The transitive closure
Transitive_relation
Copy of a directed graph with redundant edges removed
D. Equivalently, D and its transitive reduction should have the same transitive closure as each other, and the transitive reduction of D should have as
Transitive_reduction
Branch of mathematical logic
deterministic transitive closure operators yield L, problems solvable in logarithmic space. First-order logic with a transitive closure operator yields
Descriptive_complexity_theory
Logical formulation of recursion
than allow induction over arbitrary predicates, transitive closure logic allows only transitive closures to be expressed directly. FO[TC](X) is the set
Fixed-point_logic
Theory of relational databases
them is the transitive closure of a binary relation. Given a domain D, let binary relation R be a subset of D×D. The transitive closure R+ of R is the
Relational_algebra
Logic problem, AND of pairwise ORs
instance, Krom's inference rule can be interpreted as constructing the transitive closure of the graph. As Cook (1971) observes, it can also be seen as an instance
2-satisfiability
residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure operation of the latter to the former, while adding the left and right
Action_algebra
Binary relation over a set and itself
transitive relations containing R. Reflexive transitive closure, R* Defined as R* = (R+)=, the smallest preorder containing R. Reflexive transitive symmetric
Homogeneous_relation
Relationship between elements of two sets
its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal.
Binary_relation
Algorithm in graph theory
algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation, or (in connection with the Schulze voting system) widest
Floyd–Warshall_algorithm
Reflexive and transitive binary relation
a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest that preorders are almost partial
Preorder
Graph linking pairs of comparable elements in a partial order
acyclic graph, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph that has a transitive orientation, an assignment
Comparability_graph
R^{\operatorname {T} }.} Transitive closure – Smallest transitive relation containing a given binary relation Reflexive closure Franz Baader and Tobias
Symmetric_closure
X\}=\{(1,1),(1,3),(2,2),(3,3),(4,4)\}.} Symmetric closure Transitive closure – Smallest transitive relation containing a given binary relation Franz Baader
Reflexive_closure
Class of computational complexity
with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice
PSPACE
Generalized alphabetical order
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Lexicographic_order
Type of binary relation
well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel
Well-founded_relation
relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below. Ancestral relations
Ancestral_relation
Subset of incomparable elements
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Antichain
stated in terms of the transitive closure of the dependence information. Both the Omega Library and isl provide a transitive closure operation that is exact
Frameworks supporting the polyhedral model
Frameworks_supporting_the_polyhedral_model
Relationship between two sets, defined by a set of ordered pairs
its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal.
Relation_(mathematics)
Order-preserving mathematical function
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Monotonic_function
Technique for speeding up algorithms involving Boolean matrices
the Method of Four Russians may be applied include: computing the transitive closure of a graph, Boolean matrix multiplication, edit distance calculation
Method_of_Four_Russians
properties are closed under minors. closure 1. For the transitive closure of a directed graph, see transitive. 2. A closure of a directed graph is a set of
Glossary_of_graph_theory
Whether one vertex can be reached from another in a graph
{\displaystyle E} , the reachability relation of G {\displaystyle G} is the transitive closure of E {\displaystyle E} , which is to say the set of all ordered pairs
Reachability
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
Partition of vertices of a directed graph
\parallel } ), and ≍ {\displaystyle \asymp } is a transitive relation (because it is a transitive closure). As with any equivalence relation, it can be used
Weak_component
cases of more general recursive fixed-point queries, which compute transitive closures. In standard SQL:1999 hierarchical queries are implemented by way
Hierarchical and recursive queries in SQL
Hierarchical_and_recursive_queries_in_SQL
Characterizes the height of any finite partially ordered set
graph homomorphism from a given directed acyclic graph G to a k-vertex transitive tournament if and only if there does not exist a homomorphism from a (k + 1)-vertex
Mirsky's_theorem
Formal system for transcribing expressions into equivalent terms
the transitive closure of → {\displaystyle \rightarrow } . → ∗ {\displaystyle {\stackrel {*}{\rightarrow }}} is the reflexive transitive closure of →
Abstract_rewriting_system
Mathematical ranking of a set
partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two
Weak_ordering
Mathematical set with an ordering
ISBN 9781848002012. Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). "Transitive Closures of Binary Relations I". Acta Universitatis Carolinae. Mathematica
Partially_ordered_set
Divide and conquer sorting algorithm
only swapped in case their relative order has been obtained in the transitive closure of prior comparison-outcomes. Most implementations of quicksort are
Quicksort
Maximal subgraph whose vertices can reach each other
graphs may be produced as the transitive closures of arbitrary undirected graphs, for which finding the transitive closure is an equivalent formulation
Component_(graph_theory)
Type of ordering of a set
dense relation that is also transitive is said to be idempotent. Dense set — a subset of a topological space whose closure is the whole space Dense-in-itself
Dense_order
Well-quasi-ordering of finite trees
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Kruskal's_tree_theorem
Indian computer scientist
environment), Alpha (extension of relational databases with generalized transitive closure), Nest distributed system, transaction management, and database machines
Rakesh Agrawal (computer scientist)
Rakesh_Agrawal_(computer_scientist)
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
On chains and antichains in partial orders
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Dilworth's_theorem
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
Mathematical ordering with upper bounds
system ( X , → ) {\displaystyle (X,\to )} is confluent, then its transitive closure ( X , → ∗ ) {\displaystyle (X,\to ^{*})} is a directed set. Let D
Directed_set
Conversational state are the field values of a session bean plus the transitive closure of the objects reachable from the bean's fields. The name "conversational"
Conversational state (Java EE)
Conversational_state_(Java_EE)
Type of sorting algorithm that works by comparing pairs of elements
is the case when the order between a and b can be derived via the transitive closure of these prior comparison outcomes. For comparison-based sorts, the
Comparison_sort
Size of subsets in order theory
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Cofinality
Theorem of expressive equivalence between relational languages
expressible in SQL but not in relational algebra) and computing the transitive closure of a graph given by its binary edge relation (see also expressive
Codd's_theorem
Algebraic object with an ordered structure
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Ordered_field
Replacing subterm in a formula with another term
is the reflexive transitive closure of → {\displaystyle \rightarrow } . ↔ {\displaystyle \leftrightarrow } is the symmetric closure of → {\displaystyle
Rewriting
Concept in axiomatic set theory
{\displaystyle \vert y\vert <\kappa } for all y {\displaystyle y} in the transitive closure of X {\displaystyle X} forms a model of ZFC − Union {\displaystyle
Axiom_of_union
Generalization of graph theory
hypergraphs; Hall-type theorems for hypergraphs. In directed hypergraphs: transitive closure, and shortest path problems. Although hypergraphs are more difficult
Hypergraph
Relational database programming language
SQL3 Added regular expression matching, recursive queries (e.g., transitive closure), triggers, support for procedural and control-of-flow statements
SQL
Form of logic that allows quantification over predicates
set of languages definable by second-order formulas with an added transitive closure operator. EXPTIME is the set of languages definable by second-order
Second-order_logic
Mathematical proposition equivalent to the axiom of choice
every x), antisymmetric (if both x ≤ y and y ≤ x hold, then x = y), and transitive (if x ≤ y and y ≤ z then x ≤ z) is said to be (partially) ordered by ≤
Zorn's_lemma
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)
Type of monotone function
theoretically) A poset is a set equipped with a (reflexive, antisymmetric and transitive) binary relation. An order embedding A → B is an isomorphism from A to
Order_embedding
Mathematical relation inside orderings
" If a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. Such partially ordered sets are
Covering_relation
Property of rewriting systems in mathematics
of a reduction sequence from c to d. Formally, ∗→ is the reflexive-transitive closure of →. Using the example from the previous paragraph, we have (11+9)×(2+4)
Confluence (abstract rewriting)
Confluence_(abstract_rewriting)
Generalization of "n-th" to infinite cases
definition is usually justified with the transitive closure. However, the von Neumann ordinals are already transitive sets, allowing them to be formally defined
Ordinal_number
There are equally many countable order types and real numbers
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Cantor–Bernstein_theorem
and if so compute its transitive closure, in time proportional to the number of vertices and edges in the transitive closure; it remains open whether
Series-parallel_partial_order
Property of objects inherited by all their subobjects
sets. Equivalently, a set is hereditarily finite if and only if its transitive closure is finite. A hereditarily countable set is a countable set of hereditarily
Hereditary_property
Class of mathematical orderings
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Well-order
Branch of mathematics
P). Then ≤ is a 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)
Order_theory
Branch of logic
presence of a linear order, first-order logic with a commutative, transitive closure operator added yields L, problems solvable in logarithmic space. In
Finite_model_theory
Data query language developed by Facebook
language such as SPARQL, or even in dialects of SQL that support transitive closure. For example, a GraphQL interface that reports the parents of an individual
GraphQL
Axiom of set theory
from the axiom of regularity, suppose x is any set. Let t be the transitive closure of {x}. Let u be the subset of t consisting of unranked sets. If u
Axiom_of_regularity
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
Group of symmetries of an n-dimensional hypercube
Bruhat order is a natural order on every Coxeter group, defined as the transitive closure of the relation that u < w whenever there is a reflection t such that
Hyperoctahedral_group
Glossary of terms used in branch of mathematics
relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric. Adjoint. See Galois connection. Alexandrov topology
Glossary_of_order_theory
Order whose elements are all comparable
and b ≤ c {\displaystyle b\leq c} then a ≤ c {\displaystyle a\leq c} (transitive). If a ≤ b {\displaystyle a\leq b} and b ≤ a {\displaystyle b\leq a} then
Total_order
Equivalence of partially ordered sets
characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomorphism is an equivalence relation. The class of
Order_isomorphism
Alternative mathematical ordering
ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial
Cyclic_order
ISBN 978-1-118-01086-0. Lehmann, D. J. (1977). "Algebraic structures for transitive closure" (PDF). Theoretical Computer Science. 4: 59–76. doi:10.1016/0304-3975(77)90056-1
Quasiregular_element
Typographical symbol (*)
cohomology ring H ∗ ( X ) {\displaystyle H^{*}(X)} . The reflexive transitive closure of a binary relation. In statistics, z ∗ {\displaystyle z^{*}} and
Asterisk
Banach space with a compatible structure of a lattice
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Banach_lattice
Economic concept
solve this is to impose SARP, which ensures transitivity. This is characterised by taking the transitive closure of direct revealed preferences and require
Revealed_preference
Property of a relation on a set
invoked the axiom of connection: Whenever a series is originally given by a transitive asymmetrical relation, we can express connection by the condition that
Connected_relation
Bound lattice in which every element has a complement
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Complemented_lattice
Computer science concept
finite structures gains no additional power from the addition of a transitive closure operator over relations of relations (i.e., over the second-order
Polynomial_hierarchy
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
ordinals transitive 1. A transitive relation 2. The transitive closure of a set is the smallest transitive set containing it. 3. A transitive set or
Glossary_of_set_theory
Relation between state transition systems in computer science
also closed under reflexive and transitive closure; therefore, the largest simulation must be reflexive and transitive. From this follows that the largest
Simulation_(computer_science)
Special type of lattice
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Distributive_lattice
Isomorphism type of ordered sets
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Order_type
Graph algorithm
contributions. The earliest version was proposed as Paul Purdom's 1970 transitive closure algorithm, which utilized naive cycle contraction. This was optimized
Path-based strong component algorithm
Path-based_strong_component_algorithm
Assigning directions to the edges of an undirected graph
orientation. A transitive orientation is an orientation such that the resulting directed graph is its own transitive closure. The graphs with transitive orientations
Orientation_(graph_theory)
Theorem in theoretical computer science
denoted by → β {\displaystyle \rightarrow _{\beta }} and its reflexive, transitive closure by ↠ β {\displaystyle \twoheadrightarrow _{\beta }} then the Church–Rosser
Church–Rosser_theorem
Lattice formed by all integer partitions
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Young's_lattice
Equivalence of distributive lattices and set families
edges, and the initial stable sets are just the lower sets of the transitive closure of the graph. Equivalently, for a distributive lattice, the implication
Birkhoff's representation theorem
Birkhoff's_representation_theorem
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
Technique used in relational databases
nested relational algebra. extending the relational language with transitive closure, such as SQL's CONNECT statement; this allows a parent-child relation
Nested_set_model
Breadth of ideas which can be represented in a formal language
express certain types of Boolean queries, e.g. queries involving transitive closure. However, adding expressive power must be done with care: it must
Expressive power (computer science)
Expressive_power_(computer_science)
Algebraic structure used in logic
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Heyting_algebra
also includes an ILP solver based on generalized basis reduction, transitive closures on maps (which may encode infinite graphs), dependence analysis and
Integer_set_library
Concept in order theory
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Join_and_meet
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
List of Boolean algebra topics
List_of_Boolean_algebra_topics
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