### Going from Permutation Groups to Spaces-in-General

I’ve found something interesting:

Having a permutation group (in a set of permutation groups, such as the set of all (small) Euclidean spaces), we apparently can construct a space-in-general (as spaces are defined in this work):

Let $latex \pi$ be a set of permutation groups $\latex G$ (on set $latex M_G$). For spaces such as Euclidean, $M_G$ is the set of subsets of the space.

It induces ordered semigroup action with elements (spaces) $latex \mu$ determined by $latex G$ such that

$latex \langle\mu\rangle S=\{\rsupfun{f}S \mid f\in G,S\in\subsets M_G\}$, where arguments~$latex S$ are ordered by set-inclusion

and spaces~$latex \mu$ are ordered by

\$latex \mu_0\leq\mu_1\Leftrightarrow\forall G\in\pi, S\in M_G:\langle\mu_0\rangle S\leq\langle\mu_1\rangle S. \]

(TODO: Prove it is an OSA.)

(For Euclidean spaces that is the set of all moves of sets of sets.)

So, we can go from groups to spaces-in-general. Can we go in the reverse direction, making an isomorphism between groups and spaces-in-general?