Fuzzy knowledge spaces based on $\beta$ evaluation criteria

In KST, it is always assumed that the knowledge state represents items that an individual can solve in ideal conditions. Namely, the answers of individuals to items can be encoded as either correct or incorrect. The correct answer indicates a complete mastery of the item, but the incorrect answer may indicate a partial mastery of the item. It is reasonable to use a fuzzy knowledge state to represent the partial mastery of items instead of complete mastery. The fuzzy knowledge state of an individual is represented by a fuzzy set in $\mathcal{F}(Q)$ that the individual is capable of solving. For any fuzzy knowledge state, each item has a value that represents the level of individual mastery of the item. Fuzzy knowledge spaces and fuzzy learning spaces are generalizations of knowledge spaces and learning spaces. The generalization based on partial order is helpful to distinguish the equally informative items, which can directly induce a discriminative fuzzy knowledge structure. It is effective to use fuzzy knowledge spaces and fuzzy learning spaces to assess knowledge and guide further learning. A fuzzy knowledge space and a fuzzy learning space can be faithfully summarized by the fuzzy knowledge basis, since they are union-closed. Any fuzzy knowledge state of a fuzzy knowledge space can be generated by forming the union of some fuzzy knowledge states in the basis. A fuzzy knowledge basis is a generalization of the knowledge basis of a knowledge space.