| closure {relations} | R Documentation |
Computes transitive and reflexive closure of an endorelation.
transitive_closure(x)
reflexive_closure(x)
## S3 method for class 'relation'
closure(x, operation = c("transitive", "reflexive"), ...)
x |
an R object inheriting from class |
operation |
character string indicating the kind of closure. |
... |
currently not used. |
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.
S. Warshall (1962), A theorem on Boolean matrices. Journal of the ACM, 9/1, 11–12. doi: 10.1145/321105.321107.
relation(),
reflexive_reduction(),
transitive_reduction(),
closure().
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")