A Solution to the Embedding Problem for Partial Idempotent Latin Squares

A partial latin square on t symbols <rl5..., ut of side n is an n x n matrix, each of the cells of which may be empty or may be occupied by. one of the symbols ol,..., at, and which satisfies the rule that no symbol occurs more than once in any row or more than once in any column. An incomplete latin square on t symbols of side n is a partial latin square in which there are no empty cells; it is a latin square of side n if t = n. In a partial latin square of side n on t symbols a1,...,at, if cell (i,i) is occupied by symbol <r, for each i, 1 < i < n, then the partial latin square is idempotent. The object of this paper is to prove the following theorem.