Let $(S,E,\circ,\dagger,\lambda)$ be a semigroup $(S,\circ)$ with involution $f\mapsto f^\dagger$ and action $E\to E$ of element $f$ defined as $x\mapsto \lambda_f x$.
Consider the semigroup on $S\cup E$ (I assume $S$ and $E$ being disjoint) with operation $\cdot$ defined by the formulas
- $g\cdot f=g\circ f$ for $f,g\in S$;
- $x_n\cdot\dots\cdot x_0=(x_0,\dots,x_n)$ for $x_0,\dots x_n\in E$;
- $f\cdot x=\lambda_f x$ for $f\in S$, $x\in E$;
- $x\cdot f=\lambda_{f^\dagger}x$ for $f\in S$, $x\in E$;
- $f\cdot(x_0,\dots,x_n)=(f\cdot x_0,\dots,f\cdot x_n)$ for $f\in S$, $x_0,\dots x_n\in E$;
- $(x_0,\dots,x_n)\cdot f=(x_0\cdot f,\dots,x_n\cdot f)$ for $f\in S$, $x_0,\dots x_n\in E$.
Note that having the semigroup $(S\cup E,\cdot)$ and set $S$, we can reconstruct the original semigroup.
We can add an involution to $(S\cup E,\cdot)$ by having the same involution $\dagger$ on elements $f\in S$ and extending it $x^\dagger=x$ for $x\in E$ and $(x_0,\dots,x_n)^\dagger=(x_0,\dots,x_n)$.
I created the above construction in the hope that it will help my developments in general topology: For a simple example, consider a topological space in Kolmogorov's sense (a transitive and reflexive closure operator). The class of all such (Kolmogorov) closures forms a semigroup whose actions are defined by $\lambda_f=f$. It is possible (I skip the proof) to extend this semigroup to a semigroup with involution. I hope that using the above weird construction, it's possible to reduce analysis of semigroups with actions (and involution) to just analysis of semigroups with involution, so eliminating the need to study semigroup actions. This may be similar to pointfree topology, because it eliminates study of the set $E$ with remaining study of only "morphisms" in $S$.
Have you seen in literature anything similar to the above construction? Does it have a name? Any further ideas?