On the Derivation of Identities Involving Projection Functions
نویسنده
چکیده
The \unsharpness problem" is solved by the construction of a nite Although this equation fails in general, it does hold under slightly stronger hypotheses. x1 In recent years various relation-algebraic formulations of the properties of pairs of projection functions naturally associated with direct products have appeared in the literature of computer science. Two particular relevant examples are SS93, p. 162] and BZ86, p. 127]. Such a formulation is given here in the rst half of this paper. In the second half we treat some deeper issues, primarily the solution of the \unsharpness-problem" of BHSV93, p. 211]. First we formulate some standard notions of the Peirce-Schrr oder calculus of relations. Let U be the class of all sets. By a relation we mean a class of ordered pairs. Let 1 be the universal relation, that is, the class of all ordered pairs of sets, and let 1 , be the identity relation, the class of all ordered pairs of the form ha; ai. If x; y are relations, then xy is the intersection of x and y, x;y is the relative product of x and y, and x is the converse of x. A relation x is a function if x;x 1 , , and a bijection if x and x are functions. Consider two classes A and B. The direct product of A and B is AB, the class of ordered pairs ha; bi with a 2 A and b 2 B. Choose a class C A B. Let p C and q C be the projection functions from C into A and B, that is, p C is the function that maps each ordered pair in C to its rst component (in A), and q C maps each such pair to its second component (in B). Thus p C = fhha; bi; ai : ha; bi 2 Cg and q C = fhha; bi; bi : ha; bi 2 Cg, so p C (ha; bi) = a and q C (ha; bi) = b for every ha; bi 2 C. The following conditions are satissed when p = p C and q = q C :
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