Full and quarter plane complete infinite latin squares

Given any countably infinite group G there exists a sequence a1, a2, … containing each element of G exactly once such that given any g ∈ G there is a unique k with a1, a2 … ak = g. Thus any countably infinite group is sequenceable. This sequence gives rise to an infinite quarter plane latin square C with the property that, given any (gi,gj) ∈ G × G there exists a unique (r, s) ∈ Z+ × Z+ such that crs = gi and crs+1 = gj; moreover, there exists a unique (t, u) ∈ Z+ × Z+ such that ctu = gi and ct+1u = gj. Thus, C is a complete infinite latin square. A similar result is given for an infinite full plane latin square D.