A binary relation \(R\) is reflexive, iff for all \(x\) we have \(xRx\).
rel_is_reflexive(R)rel_closure_reflexive(R)
rel_reduction_reflexive(R)
an object coercible to a 0-1 (logical) square matrix, representing a binary relation on a finite set.
The rel_closure_reflexive and
rel_reduction_reflexive functions
return a logical square matrix. dimnames
of R are preserved.
On the other hand, rel_is_reflexive returns
a single logical value.
rel_is_reflexive finds out if a given binary relation
is reflexive. The function just checks whether all elements
on the diagonal of R are non-zeros,
i.e. it has \(O(n)\) time complexity,
where \(n\) is the number of rows in R.
Missing values on the diagonal may result in NA.
Reflexive closure of a binary relation \(R\),
determined by rel_closure_reflexive,
is the minimal reflexive superset \(R'\) of \(R\).
Reflexive reduction of a binary relation \(R\),
determined by rel_reduction_reflexive,
is the minimal subset \(R'\) of \(R\),
such that the reflexive closures of \(R\) and \(R'\) are equal
i.e., the largest irreflexive relation contained in \(R\).
Other binary_relations: check_comonotonicity,
pord_nd, pord_spread,
pord_weakdom, rel_graph,
rel_is_antisymmetric,
rel_is_asymmetric,
rel_is_cyclic,
rel_is_irreflexive,
rel_is_symmetric,
rel_is_total,
rel_is_transitive,
rel_reduction_hasse