And what else can be said, if so?

([Original math.SE post](http://math.stackexchange.com/questions/214125/start-with-a-topological-group-take-the-meet-of-the-two-uniformities-and-take))

In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$.  (It also has a two-sided uniformity $\mathscr{U}$, which is the join of the two.)

Now, uniformities on a given set form a complete lattice, so we can also consider the meet of the two, $\mathscr{V}$.  However, the meet of two uniformities that yield the same topology does not necessarily again yield the same topology, so it's possible that $\mathscr{T}'$, the topology coming from $\mathscr{V}$, is coarser than our original topology $\mathscr{T}$.

(Obviously, this does not happen if the group is balanced, i.e. $\mathscr{L}=\mathscr{R}$; it also does not happen if $\mathscr{T}$ is locally compact, since the meet of two uniformities yielding the same locally compact topology does again yield the same topology.  Actually, I don't know an actual case where this does happen, so I guess a first question I can ask is, are there any actual examples of this?)

So my question is, is $(G,\mathscr{T}')$ again a topological group?  Obviously inversion is continuous, since $\mathscr{V}$ makes inversion uniformly continuous, but it's not clear what would happen with multiplication.

If it is a topological group, then we can ask things like, how does $\mathscr{V}$ compare to $\mathscr{L}'$, $\mathscr{R}'$, $\mathscr{U}'$, and $\mathscr{V'}$?  (Well, obviously it's coarser than the last of these.)  And considering $\mathscr{T} \mapsto \mathscr{T}'$ as an operation on group topologies on $G$, what happens when we iterate it?  When we iterate it transfinitely?