Abstract:We derive necessary and sufficient conditions for there to exist a latin square of order n containing two subsquares of order a and b that intersect in a subsquare of order c. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order n cannot have more than \frac{m}{n}{n\choose h}/{m\choose h} subsquares of order m, where h=\lceil(m+1)/2\rceil. Indeed, the number of subsquares of order m is bounded by a polynomial of degree at most \sqrt{2m}+2 in n. (b) For all n\ge5 there exists a loop of order n in which every element can be obtained as a product of all n elements in some order and with some bracketing.
Keywords: latin square, latin subsquare, overlapping latin subsquares, full product in loops
AMS Subject Classification: 05B15 20N05