Characterizing the result of the division of fuzzy relations

The role and properties of the division operator are well known in the framework of queries addressed to regular relational databases. However, Boolean queries may turn out to be too restrictive to answer some user needs and it is desirable to consider extended queries by introducing preferences inside selection conditions. In this paper, the extension of the division operator is investigated in the context of graded relations, i.e., whose tuples are weighted. Several interpretations of the division are possible and they mainly depend on the roles of the grades attached to tuples of input relations. Their properties are examined in the perspective of a characterization of the result obtained as a quotient, similarly to that obtained for integers.