Arithmetic of a certain semigroup of probability distributions on the group $\mathbb{R}\times \mathbb{Z}(2)$

We consider a certain convolution semigroup $\mathrm{\Theta }$ of probability distributions on the group $\mathbb{R}\times \mathbb{Z}(2)$, where $\mathbb{R}$ is the group of real numbers and $\mathbb{Z}(2)$ is the additive group of the integers modulo 2. This semigroup appeared in connection with the study of a characterization problem of mathematical statistics on $a$-adic solenoids containing an element of order 2. We answer the questions that arise in the study of arithmetic of the semigroup $\mathrm{\Theta }$. Namely, we describe the class of infinitely divisible distributions, the class of indecomposable distributions, and the class of distributions which have no indecomposable factors.