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And what else can be said, if so?

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. I think it also can't happen if $G$ embeds in a locally compact group, but I didn't work out all the details there. 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?

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This has since been answered over on MathOverflow by Todd Eisworth and Julien Melleray. The meet of these two uniformities has a name, the Roelcke uniformity, and it generates the original topology. It can be described quite simply, as the uniformity generated by the entourages $\{ (x,y): x\in VyV\}$ for $V$ a neighborhood of the origin. More information can be found in the book Topological Groups and Related Structures by Arhangel'skii and Tkachenko.

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