Fuzzy relation algebras were introduced as an algebraic formalization of mathematical structures formed by fuzzy relations; these algebras are equipped with a semi-scalar product on the unit interval that is able to characterize when a relation is crisp. On the other hand, Dedekind categories serve to provide the mathematical foundations of (L)-fuzzy sets, in which the unit interval is substituted by an arbitrary complete Brouwerian lattice (L). Unfortunately, no formula in the theory of Dedekind categories expresses the fact that a given (L)-fuzzy relation is crisp. This fact justified the introduction of Goguen categories.

The goal of this paper is to study the existence of crisp counterparts of general (L)-fuzzy relations, in the sense that, if a set of equations is fulfilled by some (L)-fuzzy relations, then there also exist some crisp relations fulfilling the same set of equations. Obviously, this is an interesting and important problem, both from a theoretical and a practical point of view.

The approach chosen by the author focuses mostly on the representation theory of Goguen categories, in particular on the generation of Goguen categories by means of an abstract construction, starting with a complete Brouwerian lattice (L) and a Dedekind category (R).

The main result obtained is that, if a set of equations is fulfilled by some (L)-fuzzy relations of a Goguen category whose underlying lattice has at least a complete prime filter, then there are crisp relations fulfilling the same set of equations. The paper ends with a conjecture on the validity of the previous result, even in the case that the underlying lattice did not have any complete prime filter.