closure {relations}

Transitive and Reflexive Closure

Description

Computes transitive and reflexive closure of an endorelation.

Usage

transitive_closure(x)
reflexive_closure(x)
## S3 method for class 'relation'
closure(x, operation = c("transitive", "reflexive"), ...)

Arguments

x

an R object inheriting from class relation, representing an endorelation.

operation

character string indicating the kind of closure.

...

currently not used.

Details

Let R be an endorelation on X and n be the number of elements in X.

The transitive closure of R is the smallest transitive relation on X that contains R. The code implements Warshall's Algorithm which is of complexity O(n^3).

The reflexive closure of R is computed by setting the diagonal of the incidence matrix to 1.

References

S. Warshall (1962), A theorem on Boolean matrices. Journal of the ACM, 9/1, 11–12. doi:10.1145/321105.321107.

See Also

relation(), reflexive_reduction(), transitive_reduction(), closure().

Examples

R <- as.relation(1 : 5)
relation_incidence(R)

## transitive closure/reduction
RR <- transitive_reduction(R)
relation_incidence(RR)
R == transitive_closure(RR)

## same
require("sets")				# closure() and reduction()
R == closure(reduction(R))

## reflexive closure/reduction

RR <- reflexive_reduction(R)
relation_incidence(RR)
R == reflexive_closure(RR)
## same:
R == closure(reduction(R, "reflexive"), "reflexive")