The inductive database framework assumes that complex knowledge discovery processes can be considered as querying processes. Querying inductive databases needs for primitives to: (1) select, manipulate and query data, (2) select, manipulate and query a priori interesting patterns (i.e., the so-called inductive queries which return the patterns that satisfy some constraints), and (3) cross over patterns and data (e.g., selecting the data in which some patterns hold). In this talk, we consider the inductive query evaluation problem and the impact of the e-adequate representation concept introduced by Mannila and Toivonen. After an abstract introduction of these concepts, we will survey our contributions on condensed e-adequate representations for frequent itemsets. Not only we have studied several exact representations like the closed sets, the free sets and the disjunct-free sets but also the first approximate condensed representation of frequent itemsets that can be used in practice: the (frequent) delta-free sets. The border representation is indeed another approximate representation which is much more concise but it provides a useless approximation of pattern frequencies. The multiple uses of these representations will be sketched. It includes mining query optimization, interactive mining optimization but also different post-processing tasks that produce interesting patterns (e.g., the computation of association rules with minimal left-hand sides).