  We consider a certain convolution semigroup $\Theta$ 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 $\Theta$. Namely, we describe the class of
infinitely divisible distributions, the class of indecomposable distributions,
and the class of distributions which have no indecomposable factors.
