We take up the work of David
Geiger in Pacific J. Math., 27 (1968), 95–100 which extends the “abstract” or set
theoretic Galois theory from total to partial functions (presumably independently of
the previous work dating back to the thirties which is not mentioned). Although
Geiger claims to treat even multi-valued functions, it is clear that the basic definition
of commutativity he gives on p. 96 only makes sense for single-valued partial
functions; and although he offers some brief comments towards the end on how one
might treat an infinite base set, his results are restricted to finitary functions and
relations on a finite set. We develop his characterizations of the Galois-closed classes
in a form which is valid for the infinitary case as well. For the classes of partial
functions we obtain the infinitarily valid formulation of his criterion; for
the relations we find that we must also require closure under an additional
operation which we call “duplication”, and which is also necessary in the finite
case. Indeed, to convince oneself that Geiger’s list is incomplete* even for
finitary relations on a finite set, one need only observe that the set of finite
Cartesian powers is closed in his sense; but the smallest Galois-closed class,
i.e., preserved by all partial functions, must surely include the diagonals as
well.
