This is a follow up of [Question 163246][1]. For the reader's convenience, let me first copy&paste some basic definitions.

We let a _semimetric_ on a set $X$ be a function $d: X \times X \to [0,\infty]$ such that $d(x,x) = 0$ and $d(x,z) \le d(x,y) + d(y,z)$ for all $x,y,z \in X$. Exactly as with "classical" metrics, $d$ induces a topology on $X$, which we call the _canonical topology_ of $d$, having as a base the sets $\{y \in X: d(x,y) < r\}$ as $x$ ranges over $X$ and $r$ over $\mathbf R^+$.

Next, we take a _topological monoid_ to be, as usual, a pair $(\mathbb M, \tau)$ consisting of a (multiplicatively written) monoid $\mathbb M = (M, \cdot)$ and a topology $\tau$ on $M$ such that $\cdot$ is continuous [...] in the expected sense.

Finally, if $\mathbb M = (M, \cdot)$ is a monoid and $d$ is a semimetric on $M$, we say that $d$ is: _right $\mathbb M$-subinvariant_ if $d(xz,yz) \le d(x,y)$ for all $x,y,z \in M$; _left_ $\mathbb M$-subinvariant if it is right $\mathbb M^{\rm op}$-subinvariant, where $\mathbb M^{\rm op}$ is the dual of $\mathbb M$; and _$\mathbb M$-subinvariant_ if it is both right and left $\mathbb M$-subinvariant.

With that said, let $(\mathbb M, \tau)$ be a first-countable topological monoid, with $\mathbb M = (M, \cdot)$. It has been already observed in the [other thread][2] that $\tau$ does not need be the canonical topology of a left or right subinvariant semimetric. However, both of the answers there (my feeling is that [Chris Schommer-Pries' construction][3] can be made to work for a _suitable_ topology, even if for the moment I don't know how) do "critically" rely on the fact that $\mathbb M$ has an absorbing element. 

Based on the above, we then say that $\mathbb M$ is _resilient_ if there doesn't exist any non-trivial submonoid $S$ of $\mathbb M$ [...] with an absorbing element (in particular, this means that the only idempotent element of $\mathbb M$ is the identity).

>> **Q.** If $\mathbb M$ is resilient, is it possible for $\tau$ to be the canonical topology of a left (respectively, right) $\mathbb M$-subinvariant semimetric?

I don't think so, but it's well possible that I'm missing something trivial (it wouldn't be a novelty...). So thanks in advance for any comment, hint, pointer, or whatever.

  [1]: http://mathoverflow.net/questions/163246/if-mathbb-m-tau-is-a-topological-monoid-is-tau-always-induced-by-a-l
  [2]: http://mathoverflow.net/questions/163246/if-mathbb-m-tau-is-a-topological-monoid-is-tau-always-induced-by-a-l
  [3]: http://mathoverflow.net/a/163535/16537