Full and quarter plane complete infinite Latin squares (Q1126210)

An infinite quarter plane Latin square on a countably infinite set of elements, \(G\), is an array \(\{a_{ij}\}^\infty_{i,j=1}\) such that each row and column is a permutation of the elements in \(G\). It is complete if for any \((\alpha,\beta)\in G\times G\) there exist unique \((r,s)\) and \((t,u)\) in \(\mathbb{Z}^+\times\mathbb{Z}^+\) such that \(a_{r,s}= \alpha\), \(a_{r,s+1}=\beta\), \(a_{t,u}=\alpha\), \(a_{t+1,u}=\beta\). Let \(G\) be a countably infinite group. \textit{C. Vanden Eynden} [Discrete Math. 23, 317-318 (1978; Zbl 0392.20023)] established that \(G\) is sequenceable, that is, there exists a sequence \(a_1,a_2,\dots\) containing each element of \(G\) exactly once such that for each \(g\in G\) there exists a unique \(k\) yielding \(a_1a_2\dots a_k=g\). The author uses this result to construct a complete infinite quarter plane Latin square. He also extends these results to construct a complete infinite full plane Latin square \(\{a_{ij}\}^\infty_{i,j=-\infty}\).