Embedding a Latin square with transversal into a projective space

TL;DR

This paper characterizes how finite geometries derived from Latin squares, including those with transversals and collections of mutually orthogonal Latin squares, can be embedded into projective spaces over skew fields.

Contribution

It extends previous work by providing a characterization of embeddings of Latin square-based geometries into projective spaces over skew fields.

Findings

Finite geometries from Latin squares can be embedded into projective spaces.

Extensions of previous work on Latin square geometries are provided.

Characterizations include cases with transversals and mutually orthogonal Latin squares.

Abstract

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n^2 lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n^2-n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n^2 lines of size k. Extending work of Bruen and Colbourn (J. Combin. Th. Ser. A 92 (2000), 88-94), we characterise embeddings of these finite geometries into projective spaces over skew fields.