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. I will give a necessary and sufficient condition for an r × p (ν1, ν2, . . . , νn)-latinized rectangle to be extendible to a p × p (ν1, ν2, . . . , νn)-latinized square. The condition is a general- ization of P. Hall’s condition for the existence of a system of distinct representatives, and will be called Hall’s (ν1, ν2, . . . , νn)-Constrained Condition. I will then use this result to give two further sets of necessary and sufficient conditions.