I see two apparently different definitions for totality which don't seem to be equivalent.
Definition 1. A relation $R \subset X \times Y$ is total if it associates to every $x \in X$ at least one $y \in Y$; that is
$\forall x \in X \;\; \exists y \in Y: (x,y) \in R$.
- Gunther Schmidt and Thomas Strohlein, Relations and Groups, page 54, Springer.
- Chris Brink et al, Relational Methods for Computer Science, page 5, Springer.
Definition 1'. A (binary) relation $R$ over $X$ is total if
$\forall a, b \in X: (a,b) \in R \vee (b,a) \in R$.
Can you please elaborate on the incompatibility of these two definitions (if you think they are) and the reasons for it? Is there any de facto standard? Would you suggest other names (e.g. 'linearity' for the second property)?
Edit: More information
Looking at a number of resources on the Web, it seems that left-totality is a more common name for the first definition. For example see:
Totality seems to be a more common name for the second definition. I would personally prefer "strictly connected" for the second definition because if you call it totality, one would expect to see a connection between that and left-totality or right-totality whereas I don't think there is one.