The Product of the Families of the Groups

Let us mention that every function which is groupoid yielding is also 1-sorted yielding. Let I be a set. Note that there exists a many sorted set indexed by I which is groupoid yielding. Let us note that there exists a function which is groupoid yielding. Let I be a set. A family of semigroups indexed by I is a groupoid yielding many sorted set indexed by I. Let I be a non empty set, let F be a family of semigroups indexed by I, and let i be an element of I. Then F(i) is a non empty groupoid. Let I be a set and let F be a family of semigroups indexed by I. Note that the support of F is non-empty. Let I be a set and let F be a family of semigroups indexed by I. The functor ∏F yields a strict groupoid and is defined by the conditions (Def. 2).