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 \(f^\alpha <f^\beta \). We show that \(\mathcal {A}\) can be embedded in a disjoint continuous iteration group if and only if for any compact subinterval K of I of positive length the set \(\bigcup _{f\in \mathcal {A}}f^{-1}(K)\) contains an interval of the form (a, c). To do this we provide some results on the density of certain sets of real numbers which are crucial in establishing the main results.

令A为实数的加法半群，其生成的非循环的加法群。令\（I =（a，b）\）为一个开放区间，而\（\ mathcal {A} = \ left \ {f ^ \ alpha：\ alpha \ in A \ right \} \）为一个迭代半群固定点自由增加的功能我到我使得\（\阿尔法<\测试\）意味着\（F ^ \阿尔法<F ^ \测试\） 。我们表明，\（\ mathcal {A} \）可以被嵌入在一个不相交的连续迭代组当且仅当对于任何紧凑子区间ķ的我正长度的设定\（\ bigcup _ {F \在\ mathcal {A }} f ^ {-1}（K）\）包含形式为（a，  c）的间隔。为此，我们对某些实数集的密度提供了一些结果，这对于建立主要结果至关重要。