Categories of relations and functional relations
In this nice article the authors consider relations and their composition in a category having finite limits with \(({\mathcal E},{\mathcal M})\)-factorization structure, where \({\mathcal M}\) consists of monomorphisms but \({\mathcal E}\) is not restricted to epimorphisms [\textit{P. J. Freyd} and \textit{G. M. Kelly}, J. Pure Appl. Algebra 2, 169-191 (1972; Zbl 0257.18005)]. An associativity criterion for composition of relations is achieved [see also, \textit{A. Klein}, Ill. J. Math. 14, 536-550 (1970; Zbl 0217.07001)] and the categories of functional and induced relations are studied. It is shown that under certain assumptions, the categories of relations on functional and induced relations are isomorphic to the category of relations for the given category. Finally, an illuminating example based on the categories of fuzzy sets are taken into account. In this example functional relations are focused for genuine reason [\textit{D. Ponasse}, Fuzzy Sets Syst. 28, No. 3, 235-244 (1988; Zbl 0675.03032)].
scientific article; zbMATH DE number 1563348
scientific article