rel_is_total {agop} | R Documentation |

A binary relation *R* is *total*
(or *strong complete*), iff
for all *x*, *y* we have *xRy* or *yRx*.

rel_is_total(R) rel_closure_total_fair(R)

`R` |
an object coercible to a 0-1 (logical) square matrix, representing a binary relation on a finite set. |

Note that each total relation is also reflexive,
see `rel_is_reflexive`

.

`rel_is_total`

determines if a given binary relation
`R`

is total.
The algorithm has *O(n^2)* time complexity,
where *n* is the number of rows in `R`

.
If `R[i,j]`

and `R[j,i]`

is `NA`

for some *(i,j)*, then the functions outputs `NA`

.

The problem of finding a total closure or reduction is not well-defined in general.

When dealing with preorders, however, the following
closure may be useful, see (Gagolewski, 2013).
*Fair totalization* of *R*, performed by
`rel_closure_total_fair`

, is the minimal superset *R'* of *R*
such that if not *xRy* and not *yRx*
then *xR'y* and *yR'x*.

Even if `R`

is transitive, the resulting relation
might not necessarily fulfil this property.
If you want a total preorder,
call `rel_closure_transitive`

afterwards.
Missing values in `R`

are not allowed and result in an error.

`rel_is_total`

returns a single logical value.

`rel_closure_reflexive`

returns a logical square matrix.
`dimnames`

of `R`

are preserved.

Gagolewski M., Scientific Impact Assessment Cannot be Fair,
*Journal of Informetrics* 7(4), 2013, pp. 792-802.

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_nd`

, `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_transitive`

,
`rel_reduction_hasse`

[Package *agop* version 0.2-3 Index]