nLab
total relation

A (binary) relation $\sim$ on a set $A$ is total if any two elements that are related in one order or the other:

$$\forall (x,y:A),x\sim y\vee y\sim x$$\forall (x, y: A),\; x \sim y \;\vee\; y \sim x

In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R:A\to A$ is total if its intersection with its reverse is the universal relation:

$$A\times A\subseteq R\cup R^{\mathrm{op}}$$A \times A \subseteq R \cup R^{op}

Of course, this containment is in fact an equality.

A total relation is necessarily reflexive.

Note that an entire relation is sometimes called ‘total’, but these are unrelated concepts. The ‘total’ there is in the sense of a total (as opposed to partial) function, while the ‘total’ here is in the sense of total (as opposed to partial) order.