On the primitivity of the division of fuzzy relations

 Allowing for flexible queries enables database users to express preferences inside elementary conditions and priorities between conditions. The division is one of the algebraic operators defined in order to query regular databases. This operation aims at the selection of A-elements which are connected with (at least) a given subset of B-elements, e.g., the stores which ordered all the items supplied by a given manufacturer. It is mainly used in the framework of the relational model of data, although it makes sense in object-oriented databases as well. In the relational context, the division is a non-primitive operation which may be expressed in terms of other operations, namely projection, Cartesian product and set difference. When fuzzy predicates appear, this operator needs to be extended to fuzzy relations and this requires the replacement of the usual implication by a fuzzy one. This paper proposes two types of meaning of the extended division and it investigates the issue of the primitivity of the extended operation (i.e., if the division of fuzzy relations is expressible in terms of other operations). The final objective is to decide whether this operator is necessary or not for the purpose of flexible querying and to help the design of a query language supporting flexible queries, among which those conveying a division of fuzzy relations.