Infinite Latin Squares: Neighbor Balance and Orthogonality

TL;DR

This paper explores the properties of infinite Latin squares, demonstrating how to construct Vatican squares and sets of mutually orthogonal Latin squares for infinite groups, extending finite concepts to the infinite case.

Contribution

It introduces methods to create infinite Vatican squares and mutually orthogonal Latin squares for infinite groups, generalizing finite Latin square properties to the infinite setting.

Findings

Existence of infinite Vatican squares via permutations of Cayley tables.

Construction of sets of mutually orthogonal Latin squares for any infinite group.

Infinite groups with many square elements have strong complete mappings.

Abstract

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 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.