Deducibility and exactness
J Riguet · Logic Journal of IGPL · 1998
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 Re contained in R, in contrast with the difunctional closure of R which is larger than R. In [8] we have built from a given Ferrers relation R the relation1R [odot ] 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 [odot ] R(R-1)†R and Re are identical. It is important to notice that the construction used here for the definition of Re is made without using the Boolean difference operation. Key words: Binary relations, homology, exact squares, relational calculus, relative algebra