Well-graded polytomous knowledge structures

Heller (2021) and Stefanutti et al. (2020) provided the mathematical foundation for the generalization of knowledge structure theory (KST) to polytomous items. Based on their works, the well-gradedness can be extended to polytomous knowledge structures. We propose the concepts of discriminative polytomous knowledge structure and well-graded polytomous knowledge structure. Then we show that every well-graded polytomous knowledge structure is discriminative. The basis of any polytomous knowledge space is formed by the collection of all the atoms. We discuss the sufficient and necessary conditions of polytomous knowledge structures to be well-graded polytomous knowledge spaces. Moreover, we provide an example to illustrate that a well-graded polytomous knowledge space is not necessarily a polytomous closure space.