turkmath.org

A Latin square is an n×n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Latin squares may also be considered as multiplication tables of quasigroups. Latin squares are closely related with finite geometries and have applications in design of experiments. Here I will talk about a generalization of a theorem of M. Hall, Jr., that an r×n Latin rectangle on n symbols can be extended to an n×n Latin square on the same n symbols. Let p,n,ν_1,ν_2,...,ν_n be positive integers such that 1 ≤ ν_i ≤ p (1 ≤ i ≤ n) and ∑^n_{i=1}ν_i = p^2. Call an r × p matrix on n symbols σ_1,σ_2,...,σ_n an r × p (ν_1,ν_2,...,ν_n)-latinized rectangle if no symbol occurs more than once in any row or column, and if the symbol σi occurs at most νi times altogether (1 ≤ i ≤ n). I will give a necessary and sufficient condition for an r × p (ν_1,ν_2,...,ν_n)-latinized rectangle to be extendible to a p × p (ν_1,ν_2,...,ν_n)-latinized square. The condition is a generalization of P. Hall’s condition for the existence of a system of distinct representatives, and will be called Hall’s (ν_1,ν_2,...,ν_n)-Constrained Condition. I will then use this result to give two further sets of necessary and sufficient conditions.