If (S(n, m))(n, m) ⩾ (0, 0) is a multiplicative family of partial isometries on \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathbb {Z}^{2}$\end{document} with the lexicographic order, then S(1, 0) and S(0, 1) commute in a certain weak sense. Let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\widetilde{A}$\end{document} and be commuting unitary extensions of S(1, 0) and S(0, 1). We give a sufficient condition for \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\big (\widetilde{A}^{n} \widetilde{B}^{m}\big )_{(n,m) \in \mathbb {Z}^{2}}$\end{document} to be a unitary extension of the given family. Under this condition we present a description of the set of extensions. We also describe the set of all minimal commuting unitary extensions of any pair of partial isometries that commutes in the same weak sense that S(1, 0) and S(0, 1) commute. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim