Concept lattice is a partial order set of concept elements, which can be used to analyze and process the complete information system described by a formal context. However, in most cases, the information system is incomplete. The rough set theory is an effective mathematical tool to deal with incompleteness and uncertainty. From the perspective of rough sets, an augmented formal context is defined to describe an incomplete information system. Then, the extreme concept and the extreme concept lattice are respectively defined on the basis of the concept lattice theory, and their properties are discussed. Furthermore, an extreme concept generating algorithm from the augmented formal context is proposed. In order to obtain a more concise decision rule, a new decision rule obtaining algorithm without redundant attributes is also proposed. Finally, calculation example and comparition of UCI dataset verify the feasibility and effectiveness of the proposed algorithm, particularly, when the incomplete information system becomes complete, the extreme concept degenerates into the classical concept.