The phenomemnon of minimality is well-studied in the realm of topological groups.
Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a topological group $Y$ is a topological isomorphism.
I am interested if the notion of minimality was studied in the realm of semi-topological or quasi-topological groups.
A semi-topological group is a group $G$ endowed with a Hausdorff topology making the group operation $G\times G\to G$, $(x,y)\mapsto xy$, separately continuous.
A semitopological group is called a quasi-topological group if the operation of inversion $G\to G$, $x\mapsto x^{-1}$, is continuous.
Let us define a semi-topological group $X$ to be semi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a semitopological group $Y$ is a topological isomorphism.
By analogy we can define a quasi-topological group $X$ to be quasi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a quasi-topological group $Y$ is a topological isomorphism.
It is clear that each semi-minimal quasi-topological group is quasi-minimal and each quasi-minimal topological group is minimal.
It can be shown that each compact Hausdorff semitopological group is semi-minimal (topological group).
Problem 1. Is each semi-minimal semi-topological group compact?
Problem 2. Is each quasi-minimal quasi-topological group compact?
I even cannot prove or disprove the following
Conjecture. No countable Boolean semi-topological group is semi-minimal.
Remark. By the answer to this MO problem, for some submonoid $M$ of the monoid $\omega^\omega$ of self-maps of a countable set $X$ there is no minimal Hausdorff topology on $\omega$ in which all self-maps $f\in M$ are continuous. At the moment this is the unique (known to me) example of an algebraic systems with unary operations, admitting no minimal Hausdorff topology.
