Infinite Latin Squares: Neighbor Balance and Orthogonality

Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $\mathrm{\mid }G\mathrm{\mid }$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $\mathrm{\mid }G\mathrm{\mid }$ mutually orthogonal orthomorphisms and hence there is a set of $\mathrm{\mid }G\mathrm{\mid }$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $\mathrm{\mid }G\mathrm{\mid }$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.