Infinite Latin Squares: Neighbor Balance and Orthogonality [article]

Anthony B. Evans, Gage N. Martin, Kaethe Minden, M. A. Ollis
2018 arXiv   pre-print
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 |G|-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
more » ... show that every infinite group G has a set of |G| mutually orthogonal orthomorphisms and hence there is a set of |G| mutually orthogonal Latin squares based on G. We show that an infinite group G with |G|-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.
arXiv:1807.08580v1 fatcat:rh5rwsymhrgndlaoauqr6fdkg4