  Infinite products, indexed by countably infinite linear orders, arise
naturally in the context of fundamental groupoids. Such products are called
"transfinite" if the index orders are permitted to contain a dense suborder and
are called "scattered" otherwise. In this paper, we prove several technical
lemmas related to the reduction (i.e. combining of factors) of transfinite
products in fundamental groupoids. Applying these results, we show that if the
transfinite fundamental group operations are well-defined in a space $X$ with a
scattered algebraic $1$-wild set $\mathbf{aw}(X)$, then all transfinite
fundamental groupoid operations are also well-defined.
