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 $\mathbf R^+$. We say that a topology $\tau$ on $X$ is _semimetrizable_ if there exists a semimetric $d$ on $X$ such that $\tau$ is the canonical topology of $d$.

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][1] that $\tau$ does not need be semimetrizable (see Chris Schommer-Pries' construction [here][2] and the comments to his answer). Furthermore, $\tau$ doesn't need be induced by a left or right subinvariant semimetric even in the case when the topology is metrizable (see [here][3]); however, the construction used for proving it does "critically" depend 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). Then here are my questions:

>> **Q1.** If $\mathbb M$ is resilient and $\tau$ is [semi]metrizable, is it true that $\tau$ is the canonical topology of a left (respectively, right) $\mathbb M$-subinvariant semimetric? **Q2.** And what about the case when $\mathbb M$ is, in addition, cancellative (i.e., $xz=yz$ or $zx=zy$ for some $x,y,z \in M$ imply $x=y$)?

Thanks in advance for any comment, hint, pointer, or whatever. (Some parts of the current formulation of the OP are based on comments by Eric Wofsey and Chris Schommer-Pries.)


  [1]: https://mathoverflow.net/questions/163246/if-mathbb-m-tau-is-a-topological-monoid-is-tau-always-induced-by-a-l
  [2]: https://mathoverflow.net/a/163535/16537
  [3]: https://mathoverflow.net/a/163531/16537