Point differentiations of algebras of generalized analytical functions

For any natural $k$ (which can also be $\infty$) a quasimaximal semi-group $\Gamma_0$ has been constructed on the half axis, such that the space of point differentiations in non-idempotent characters of $\Gamma_0$ of the algebra $A(\Gamma_0)$ of generalized analytical functions be $k$-dimensional.