Quasi-embeddings and intersections of latin squares of different orders

We consider a common generalization of the embedding and intersection problems for latin squares. Specifically, we extend the definition of embedding to squares whose sides do not meet the necessary condition for embedding and extend the intersection problem to squares of different orders. Results are given for arbitrary latin squares, and those which are idempotent and idempotent symmetric. The latter topic is interpreted in terms of one-factorizations of the complete graph. Similar problems for Steiner triple systems, which can be regarded as totally symmetric idempotent latin squares, had been previously investigated by the authors, along with P. Danziger and T. Griggs.