Completing a solution of the embedding problem for incomplete idempotent latin squares when numerical conditions suffice

Building upon the work of several previous papers, necessary and sufficient conditions are obtained for an incomplete idempotent latin square R of order n to be embedded in an idempotent latin square of order 2 n . This work extends known necessary and sufficient conditions for embeddings into idempotent latin squares of order t ≥ 2 n + 1 , thereby pushing the bounds down to the point where subsequent developments must handle additional conditions that involve the arrangement of symbols on the given square.