Compatibility of fuzzy power relations

In this paper we study the conditions under which the compatibility of binary fuzzy relations is preserved by the operation of powering. We pay particular attention to fuzzy power algebras based on the cartesian product of fuzzy sets. We show that the preservation of compatibility essentially depends on the underlying structure of truth values LL. Among other things we prove that for fuzzy preorders both Hoare-goodness and Smyth-goodness imply compatibility, but the converse is generally true only under the assumption that LL is a complete Heyting algebra.