Vol. 79, No. 1, 1978

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The Galois connection between partial functions and relations

Isidor Fleischer and Ivo G. Rosenberg

Vol. 79 (1978), No. 1, 93–97

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.

Mathematical Subject Classification 2000
Primary: 08A55
Received: 23 May 1977
Published: 1 November 1978
Isidor Fleischer
Ivo G. Rosenberg