In polytomous KST, each item can be assigned a different response scale, and the individual's answer to items reflects the individual's understanding level of knowledge. The polytomous knowledge space can be described by its basis, and the polytomous closure space can be represented by its minimal intersection generation group. The basis of a polytomous knowledge space based on different response scales is composed of its atoms, while the minimal intersection generation group of a polytomous closure space is composed of all intersection generators. Based on polytomous knowledge structures of different response scales of items, the polytomous knowledge state of an individual can be evaluated in the finite steps of item tests. Based on polytomous knowledge structures of different response scales of items, the polytomous knowledge state of an individual can be evaluated by testing some items so as to guide learning of individuals.

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.