On the composition of fuzzy power relations

This paper deals with the “powering” of binary fuzzy relations (i.e. lifting a fuzzy relation R defined on a set X to the relation R+ defined on the set F(X) of all fuzzy subsets of X). We prove that for any complete residuated lattice L, the composition of the powers of two L-relations is always a subset of the power of their composition. Answering to a question posed by Georgescu, we prove that the converse is not always true. We prove that the composition of the powers of two L-relations is equal to the power of their composition if and only if L is a Heyting algebra.