Smallest large or largest small

The greatest number that can be in both sets is $x=91$ . One way to achieve this is to put the numbers from 91, 92, ...100 in the first column. Then Set A = {91, 92, ...100} and Set B includes 91 from that first column. Clearly, this is the minimum, as Set B can never include a number more than 91 because 92 can never be the minimum in a row of 10 numbers.

The two sets can never have more than $y=1$ number in common. To see this, assume that there is one number in common, and call it N. Then every number in N's column is greater than or equal to N, so no number less than N can be the maximum in its row, and therefore no number less than N can be in set A. Similarly, no number greater than N can be in set B. Therefore, N is the only number that the sets have in common.

The product sought is then $xy=91*1=\boxed{91}$