Maximal Representable Subfamilies

A family F = (F(i) \iel) is a function from an index set / into the set of all non-empty subsets of a set. A subset G e F is called a subfamily of F and a function / f rom / into u (F(/) \iel} such that / ( t ) e F(i) is called a choice function. Let F be a function. Dmn F denotes the domain of F and rng F denotes the range of JF. If F is a function and J £ dmn F, then F \ J will denote the restriction of F to J. If i / is a subfamily of F then it is sometimes useful to write FH for the family FII = {(U F ( 0 \ u rng//) | F(Q $ u rngH}. Let IA(F) be the set of all injective choice functions of F. A subfamily G of F is called critical if L4(G) ^ 0 and if for every/eL4(G) we have that r n g / = u rngG. Lemma 1 of [5] implies that every F has a maximal critical subfamily, which may be empty. The connection between injective choice functions (i.c.f.) and critical families is shown by the following lemma.