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...

{\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...

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...

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...

without urelements is transitive. The transitive closure of a set X {\displaystyle X} is the smallest (with respect to inclusion) transitive set that includes...

deterministic transitive closure operators yield L, problems solvable in logarithmic space. First-order logic with a transitive closure operator yields...

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...

than allow induction over arbitrary predicates, transitive closure logic allows only transitive closures to be expressed directly. FO[TC](X) is the set...

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...

various operations such as outer joins, aggregate functions and even transitive closure. Whereas the result of a join (or inner join) consists of tuples formed...

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....

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....

algorithm. Versions of the algorithm can also be used for finding the transitive closure of a relation R {\displaystyle R} , or (in connection with the Schulze...

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...

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...

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...

with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice...

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...

partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two...

relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric. Adjoint. See Galois connection. Alexandrov topology...

transitive relations containing R. Reflexive transitive closure, R* Defined as R* = (R+)=, the smallest preorder containing R. Reflexive transitive symmetric...

important; while looking at the transitive closure of a system (all nodes downstream from a node), a node in its own transitive closure indicates a circularity;...

acyclic graph, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph that has a transitive orientation, an assignment...

is the reflexive transitive closure of → {\displaystyle \rightarrow } . ↔ {\displaystyle \leftrightarrow } is the symmetric closure of → {\displaystyle...

the Method of Four Russians may be applied include: computing the transitive closure of a graph, Boolean matrix multiplication, edit distance calculation...

relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below. Ancestral relations...

extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology...

graphs may be produced as the transitive closures of arbitrary undirected graphs, for which finding the transitive closure is an equivalent formulation...

itself, are examples of closure operators since they satisfy all of the Kuratowski closure axioms. As a result, the upper closure of a set is equal to the...

only swapped in case their relative order has been obtained in the transitive closure of prior comparison-outcomes. Most implementations of quicksort are...