The full n-Latin square is the n × n array with symbols 1, 2, . . . , n in each cell. In a way that is analogous to critical sets of full designs, a critical set of the full n-Latin square can be used to find a defining set for any Latin square of order n. In this paper we study the size of the smallest critical set for a full n-Latin square, showing this to be somewhere between (n 3 − 2n 2 + 2n)/2 and (n − 1) 3 + 1. In the case that each cell is either full or empty, we show the size of a critical set in the full n-Latin square is always equal to n 3 − 2n 2 − n.

An (m, n, 2)-balanced Latin rectangle is an m × n array on symbols 0 and 1 such that each symbol occurs n times in each row and m times in each column, with each cell containing either two 0's, two 1's or both 0 and 1. We completely determine the structure of all critical sets of the full (m, n, 2)-balanced Latin rectangle (which contains 0 and 1 in each cell). If m, n ≥ 2, the minimum size for such a structure is shown to be (m − 1)(n − 1) + 1. Such critical sets in turn determine defining sets for (0, 1)-matrices.