total relation

In mathematics, a binary relation ''R'' ⊆ ''X''×''Y'' between two sets ''X'' and ''Y'' is total (or left total) if the source set ''X'' equals the domain . Conversely, ''R'' is called right total if ''Y'' equals the range .

When ''f'': ''X'' → ''Y'' is a function, the domain of ''f'' is all of ''X'', hence ''f'' is a total relation. On the other hand, if ''f'' is a partial function, then the domain may be a proper subset of ''X'', in which case ''f'' is not a total relation.

"A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."Functions
from Carnegie Mellon University

Algebraic characterization

Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let $X,Y$ be two sets, and let $R\subseteq X\times Y.$ For any two sets $A,B,$ let $L_=A\times B$ be the universal relation between $A$ and $B,$ and let $I_A=\$ be the identity relation on $A.$ We use the notation $R^\top$ for the converse relation of $R.$

* $R$ is total iff for any set $W$ and any $S\subseteq W\times X,$ $S\ne\emptyset$ implies $SR\ne\emptyset.$
* $R$ is total iff $I_X\subseteq RR^\top.$
* If $R$ is total, then $L_=RL_.$ The converse is true if $Y\ne\emptyset.$If $Y=\emptyset\ne X,$ then $R$ will be not total.
* If $R$ is total, then $\overline=\emptyset.$ The converse is true if $Y\ne\emptyset.$Observe $\overline=\emptyset\Leftrightarrow RL_=L_,$ and apply the previous bullet.
* If $R$ is total, then $\overline R\subseteq R\overline.$ The converse is true if $Y\ne\emptyset.$ Definition 5.8, page 57.
* More generally, if $R$ is total, then for any set $Z$ and any $S\subseteq Y\times Z,$ $\overline\subseteq R\overline S.$ The converse is true if $Y\ne\emptyset.$Take $Z=Y,S=I_Y$ and appeal to the previous bullet.

Notes

References

* Gunther Schmidt & Michael Winter (2018) ''Relational Topology''
* C. Brink, W. Kahl, and G. Schmidt (1997) ''Relational Methods in Computer Science'', Advances in Computer Science, page 5,
* Gunther Schmidt & Thomas Strohlein (2012) 987
* Gunther Schmidt (2011)

{{Order theory

Binary relations