Normalization of fuzzy algebras

In this work the notion of fuzzy algebra is investigated. Given an algebra and a poset S, a fuzzy algebra is an S-set f over the base set A of the given algebra such that every cut, that is the set of all x in A such that f(x) is greater or equal to a, where a is any element of S, is the base set of a subalgebra of the given algebra. Some consequences are derived. A particular notion of equivalence between two fuzzy algebras is studied, which is more perspicuous than the notion of isomorphism. A set of properties which appear to be desirable for a fuzzy algebra is settled. The main theorem of the paper says that every fuzzy algebra is 'normalizable', i.e. another fuzzy algebra may be constructed which is equivalent to the former and which satisfies all the desired properties.