§ Algebraic structure for vector clocks

I update my time, ie, union(time me, time me), I get an element that's one up the lattice. When I union with someone else, I get the max. So we have an algebraic structure which is $(L, \leq, next: L \rightarrow L)$ where next is monotone for (L, <=). The induced union operator $\cup: L \times L \rightarrow L$ is:

$$x \cup y \equiv \begin{cases} \texttt{next}(x) & x = y \\ \max(x, y) & x \neq y \end{cases}$$

• Is the total ordering on vector clocks not isomorphic to the total ordering on $\mathbb{R}$?

This also shows up in the case of the "Gallager-Humblet-Spira Algorithm" and the fragment-name-union-rule.