pord_nd {agop} R Documentation

## Weak Dominance Relation (Preorder)

### Description

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

### Usage

pord_nd(x, y, incompatible_lengths = NA)


### Arguments

 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

### Details

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

1. n_x=n_y and

2. for all i=1,\dots,n_x it holds x_i\le 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.

### Value

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

### References

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()