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,…,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.

### 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`