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.
Highlights
A finite Latin square is row complete or Roman if any two distinct symbols appear in adjacent cells within rows once in each order
Finite row complete squares exist for all composite orders [18] and finite complete squares are known to exist for all even orders [15] and many odd composite orders at which a nonabelian group exists; see, for example, [22]
We show that there is a Vatican square of each infinite order that cannot be produced by permuting the rows and columns of a Cayley table
Summary
A finite Latin square is row complete or Roman if any two distinct symbols appear in adjacent cells within rows once in each order. Say that an infinite Latin square on index set I is semi-Vatican if for each d with I(d) = ∅ we have that each pair of distinct symbols appears at distance d exactly once in rows and once in columns. This does not have a finite analogue, all known constructions for finite Vatican squares of even order n have n/2 rows that together form a “row semi-Vatican rectangle” and the remaining n/2 rows are the reverse of these ones. We construct orthomorphisms directly, finding κ mutually orthogonal orthomorphisms for each group of infinite order κ and show that many infinite groups have strong complete mappings
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