Previously studied fuzzy bipolar conditions of type "and if possible" are made of a mandatory condition c and an optional condition w. They allow expressing complex preferences of a conjunctive nature. We define in this paper, a new kind of fuzzy bipolar conditions of the form "or else" which express complex preferences of a disjunctive nature. We show that the "or else" form can be used as a negation operator of the "and if possible" form and vice versa. We also show that these both forms are compatible and, therefore, fuzzy bipolar conditions of both types can be used together in the same bipolar query.

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The expression of user preferences into queries addressed to a relational database management system allows to obtain more appropriate answers. In this context, it is possible to consider fuzzy bipolar conditions which are made of two parts: i) a fuzzy constraint to represent a mandatory requirement, ii) a fuzzy wish to represent an optional requirement. Fuzzy constraints and fuzzy wishes express user preferences and are defined by fuzzy sets. This article is a first attempt to extend the relational algebra and the SQL language to bipolar fuzzy conditions.

A fuzzy bipolar relation is a relation defined by a fuzzy bipolar condition, which could be interpreted as an association of a constraint and a wish. In this context, the extension of the relational division operation to bipolarity is studied in this paper. Firstly, we define a bipolar division when the involved relations are crisp. Then, we define, from the semantic point of view, several forms of bipolar division when the involved relations are defined by fuzzy bipolar conditions. These forms of division can be related to the general model of fuzzy bipolar division introduced in [4,5].