Can totality be defined in terms of left-totality (or right-totality)?

Question

In the context of relations, can totality be defined in terms of left-totality (or right-totality)?

I ask this because both properties have the word "totality" in their names and one may think that there is a logical connection between them.

Answer 1

Totality can't be defined in terms of left-totality and right-totality. Even if a relation $R$ on a set $A$ is both left-total and right-total it might not be total.

Example: $A = \{1,2\}$ and $R=\{(1,2),\;(2,1)\}$.

$R$ is left-total and right-total. But it is not total since $(1,1)\notin R$.

Totality implies both left-totality and right-totality because in a total relation, any $x,y$ in the set are related. One could argue either way whether or not this relationship is impled by the similar naming. Although the connotation of "total" in each case is similar, I don't think it's strong enough to read too much into it.

Definitions for these terms are within these links: Special types of binary relations and Relations over a set.