pord_nd: Weak Dominance Relation (Preorder)
In agop: Aggregation Operators and Preordered Sets

Description Usage Arguments Details Value References See Also

Checks whether a numeric vector of arbitrary length is (weakly) dominated (elementwise) by another vector of the same length.

1	pord_nd(x, y, incompatible_lengths = NA)

`x`	numeric vector with nonnegative elements
`y`	numeric vector with nonnegative elements
`incompatible_lengths`	single logical value, value to return iff lengths of `x` and `y` differ

We say that a numeric vector x of length n_x is weakly dominated by y of length n_y iff

n_x=n_y and
for all i=1,…,n_x it holds x_i≤ y_i.

This relation is a preorder: it is reflexive (see rel_is_reflexive) and transitive (see rel_is_transitive), but not necessarily total (see rel_is_total). See rel_graph for a convenient function to calculate the relationship between all pairs of elements of a given set.

Such a preorder is tightly related to classical aggregation functions: each aggregation function is a morphism between weak-dominance-preordered set of vectors and the set of reals equipped with standard linear ordering.

Returns a single logical value indicating whether x is weakly dominated by y.

Grabisch M., Marichal J.-L., Mesiar R., Pap E., Aggregation functions, Cambridge University Press, 2009.

Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7

Other binary_relations: check_comonotonicity, pord_spread, pord_weakdom, rel_graph, rel_is_antisymmetric, rel_is_asymmetric, rel_is_cyclic, rel_is_irreflexive, rel_is_reflexive, rel_is_symmetric, rel_is_total, rel_is_transitive, rel_reduction_hasse