# Anything referenced from a reachable object is itself reachable ; more formally, reachability is a transitive closure.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

Here represents the reflexive and transitive closure of the step relation meaning any number of consecutive steps ( zero, one or more ).

A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator.

A full transitive closure is not needed ; a commutative transitive closure and even weaker forms suffice.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

More precisely, it is the transitive closure of the relation " is the mother of ".

However, many different DAGs may give rise to this same reachability relation: for example, the DAG with two edges a → b and b → c has the same reachability as the graph with three edges a → b, b → c, and a → c. If G is a DAG, its transitive reduction is the graph with the fewest edges that represents the same reachability as G, and its transitive closure is the graph with the most edges that represents the same reachability.

The transitive closure of G has an edge u → v for every related pair u ≤ v of distinct elements in the reachability relation of G, and may therefore be thought of as a direct translation of the reachability relation ≤ into graph-theoretic terms: every partially ordered set may be translated into a DAG in this way.

The smallest transitive set containing a set A is called the transitive closure of A.

In computer science, the Floyd – Warshall algorithm ( also known as Floyd's algorithm, Roy – Warshall algorithm, Roy – Floyd algorithm, or the WFI algorithm ) is a graph analysis algorithm for finding shortest paths in a weighted graph ( with positive or negative edge weights ) and also for finding transitive closure of a relation R. A single execution of the algorithm will find the lengths ( summed weights ) of the shortest paths between all pairs of vertices, though it does not return details of the paths themselves.

However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 and also by Stephen Warshall in 1962 for finding the transitive closure of a graph.

However, the transitive closure of set membership for such hypergraphs does induce a partial order, and " flattens " the hypergraph into a partially ordered set.

In particular, there is no transitive closure of set membership for such hypergraphs.

* Proves that the Schwartz set is the set of undominated elements of the transitive closure of the pairwise preference relation.

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R < sup >+</ sup > on set X such that R < sup >+</ sup > contains R and R < sup >+</ sup > is minimal ( Lidl and Pilz 1998: 337 ).

If the binary relation itself is transitive, then the transitive closure is that same binary relation ; otherwise, the transitive closure is a different relation.

* direct or absolutive case, marked with the bare root ; this indicates the subject of an intransitive sentence and the direct object of a transitive sentence ( a man )

* oblique-ergative case, marked in -; this indicates either the subject of a transitive sentence, targets of preverbs, or indirect objects which do not take any other suffixes ( ( to ) a child )

* 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 ;

* while looking at the transitive closure of a system, subsumption between pairs of rows indicates redundancy ;

* It is transitive: If there is a path from u to v and a path from v to w, the two paths may be concatenated together to form a path from u to w.

* The empty relation R on a non-empty set X ( i. e. aRb is never true ) is vacuously symmetric and transitive, but not reflexive.

Although siblinghood is symmetric ( if A is a sibling of B, then B is a sibling of A ) and transitive on any 3 distinct people ( if A is a sibling of B and C is a sibling of B, then A is a sibling of C, provided A is not C ( Note that " is a sibling of " is NOT a transitive relation, since A R B, and B R A implies A R A by transitivity )), it is not reflexive ( A cannot be a sibling of A ).

* Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0.

Hence R is transitive.

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder ( weak order ), an equivalence relation, or a relation with any other special properties, if and only if R is.

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.

A non-prime attribute of R is an attribute that does not belong to any candidate key of R. A transitive dependency is a functional dependency in which X → Z ( X determines Z ) indirectly, by virtue of X → Y and Y → Z ( where it is not the case that Y → X ).

A set with a binary relation R on its elements that is reflexive ( for all a in the set, aRa ), antisymmetric ( if aRb and bRa, then a = b ) and transitive ( if aRb and bRc, then aRc ) is described as a partially ordered set or poset.

* A transitive set is a set A such that whenever x ∈ A, and y ∈ x, then y ∈ A.

For example, if X is a set of airports and x R y means " there is a direct flight from airport x to airport y ", then the transitive closure of R on X is the relation R < sup >+</ sup >: “ it is possible to fly from x to y in one or more flights .”

A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z.

Examples of transitive relations include the equality 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.

The transitive closure of this relation is a different relation, namely " there is a sequence of direct flights that begins at city x and ends at city y ".

In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x.

There is a subtle cardinal ≤ κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠

This action of GL < sub > k </ sub >( R ) on F < sub > x </ sub > is both free and transitive ( This follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another ).

# If x is an element of U and if y is an element of x, then y is also an element of U. ( U is a transitive set.

* transitive if for any two x, y in X there exists a g in G such that g · x

* regular ( or simply transitive ) if it is both transitive and free ; this is equivalent to saying that for any twox, y in X there exists precisely one g in G such that g · x =