The principal result of the paper is that, if
r
>
ω
r\, > \,\omega
and
{
A
i
}
i
>
r
{\{ {A_i}\} _{i > r}}
is a partition of
ω
\omega
, then there exist
i
>
r
i\, > \,r
and infinite subsets B and C of
ω
\omega
such that
∑
F
∈
A
i
\sum F\, \in \,{A_i}
and
∏
G
∈
A
i
\prod {G\, \in \,{A_i}}
whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.