The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup pgr of convex characteristic functions on the semiaxis $(0,\infty)$. $N(\pi)=\{\varphi\in\pi:\varphi_1\mid\varphi\Rightarrow\varphi_1\equiv1\text{ or }\varphi_1=\varphi\}$ and $I_0(\pi)=\{\varphi\in\pi:\varphi_1\mid\varphi\Rightarrow\varphi_1\notin N(\pi)\}$. The following results are proved: 1) The semigroup pgr is almost delphic in the sense of R. Davidson. 2) $N(\pi)$ is a set of the type $G_\delta$ which is dense in $\pi$ (in the topology of uniform convergence on compacta). 3) The class $I_0(\pi)$ contains only the function identically equal to one.