The Non-commutative Specker Phenomenon in the Uncountable Case Saharon Shelah and Katsuya Eda

An infinitary version of the notion of free products has been introduced and investigated by G. Higman [5]. Let Gi(i ∈ I) be groups and ∗i∈XGi the free product of Gi(i ∈ X) for X ⋐ I and pXY : ∗i∈YGi → ∗i∈XGi the canonical homomorphism for X ⊆ Y ⋐ I. ( X ⋐ I denotes that X is a finite subset of I.) Then, the unrestricted free product is the inverse limit lim ←−(∗i∈XGi, pXY : X ⊆ Y ⋐ I). We remark ∗i∈∅Gi = {e}. For the simplicity, we abreviate lim ←−(∗i∈XGi, pXY : X ⊆ Y ⋐ I) and lim ←−(∗n∈ωZn, pmn : m ≤ n < ω) by lim ←−∗Gi and lim ←−∗Zn in the sequel. For S ⊆ I, pS : lim ←−∗Gi → lim ←−∗Gi be the canonical projection defined by: pS(x)(X) = x(X ∩ S) for X ⋐ I.