  In 1991, Shalaby conjectured that any $\mathbb{Z}_{n}$, where $n\equiv 1$ or
$3\pmod{8},\ n\ge 11$, admits a strong Skolem starter. In 2018, the authors
explicitly constructed some infinite "cardioidal" families of strong Skolem
starters. No other infinite families of these combinatorial designs were known
to date.
  Statements regarding the products of starters, proven in this paper give a
new way of generating strong or skew Skolem starters of composite orders. This
approach extends our previous result by generating new infinite families that
are not cardioidal.
  The products that we introduce in this paper are multi-valued binary
operations which produce 2-partitions of the set $\mathbb{Z}^*_{nm}$ of
integers modulo $nm$ without zero, from a pair of 2-partitions of
$\mathbb{Z}^*_n$ and $\mathbb{Z}^*_m$, where $n, m\ge 3$ are odd integers. We
prove several remarkable properties of these operations applied to starters in
$\mathbb{Z}_n$ and to some other combinatorial objects that are 2-partitions of
$\mathbb{Z}^*_n$ with additional restrictions, such as strong, skew, Skolem and
cardioidal 2-partitions.
