Notes on the polytomous generalization of knowledge space theory

Stefanutti et al. (2020) and Heller (2021) have recently done significant work on the polytomous extensions of knowledge space theory (KST), which opens the field for systematically generalizing many KST concepts to the polytomous case. Following these developments, the paper provides a first counterexample showing that the assumptions in Heller (2021) do not guarantee component-directed joins to be defined item-wise. This leads to an incomplete characterization of the closed elements of the Galois connection in Proposition 8 of Heller (2021), an issue which is resolved in the present paper. A second counterexample in the paper shows that the equivalence between atoms and $⨆$-irreducible elements of the polytomous structure stated in Stefanutti et al. (2020) may not hold in general. This paper provides theoretical results showing that the equivalence still holds if the response categories form a linear order or the structure happens to be factorial.