Certain density theorems applied to the embeddability of iteration semigroups

Let A be an additive semigroup of real numbers the additive group generated by which is non-cyclic. Let $$I=(a,b)$$ be an open interval and $$\mathcal {A}=\left\{ f^\alpha :\alpha \in A\right\} $$ be an iteration semigroup of fixed point free increasing functions of I onto I such that $$\alpha <\beta $$ implies $$f^\alpha