One-transversal families

Extract

Let I be a set and

be a family of sets A_ν labelled by the elements of I. Throughout, parentheses ( ) denote families and curly brackets {} denote sets. A transversal, or system of distinct representatives, of F is, by definition, a family (x_ν: ν ∈ I) of objects x_ν, for ν ∈ I, such that †

Thus the family ({1, 2,} {1, 2}) has exactly two transversals, namely (1,2) and (2, 1). Let TF denote the set of all transversals of F. Much work has been done on the question of characterizing those F for which TF ≠ Ø. We are here going to characterize those F for which TF has exactly one element. In contrast to the more familiar case TF ≠ Ø, our argument is effective irrespective of the cardinalities of I and A_ν.