Deducibility and Exactness

What I intend to show in this short paper is how one can translate in relational terms the concepts of deducibility and exactness which are the result of a sequence of works on homology theory and algebraic topology. As we shall see, we shall obtain as a final product the possibility to associate to an arbitrary binary relation R a difunctional relation R contained in R, in contrast with the difunctional closure of R which is larger that R. In [8] we have built from a given Ferrers relation R the relation1 R ⊙ R ( R−1 † R and proved its difunctionality, but in fact, as already noticed by Schmidt and Ströhlein ([10] p. 78 Prop. 4.4.14) R ⊙ R ( R−1 † R is difunctional even when R is arbitrary. I shall show that, in fact, R ⊙ R ( R−1 † R and Re are identical. It is important to notice that the construction used here for the definition of R is made without using the Boolean difference operation.