There are mainly two classes of approaches in the studies of Formal Concept Analysis (FCA), i.e. the constructive and axiomatic approaches. In axiomatic approach, operators are interpreted by using operations in mathematical systems instead of operations in a formal context. Seeking for minimal axioms to characterize the concept generation operators is an important issue in the research of the axiomatic approach. In this paper, axiomatic characterizations of set-theoretic operators are investigated. We construct an adjoint generalized (dual) concept systems in which the pair of classical concept generation operators are represented by one set-theoretic operator, and the other operator can be obtained from the former. Compared with the previous methods, the proposed generalized (dual) concept systems have fewer axioms and is easy to verify. Some properties of adjoint generalized (dual) concept systems are examined.
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