Dedekind categories with cutoff operators (Q549312)

This paper is concerned with \textit{crispness} of \(L\)-fuzzy relations. It is known that the category of \(L\)-fuzzy relations is a Dedekind category. Since there is no formula in a first-order relational language that characterizes crispness in a Dedekind category [\textit{M. Winter}, Inf. Sci. 139, No.~3--4, 233--252 (2001; Zbl 0992.18004)], so, the authors introduce cutoff operators in this paper to study crispness in Dedekind categories. Cutoff operators are, intuitively, generalizations of the operator that maps a \(L\)-fuzzy relation to the largest crisp relation it contains. A representation theorem is obtained for Dedekind categories with a cutoff operator satisfying the point axiom.