CD-polytomous knowledge spaces and corresponding polytomous surmise systems

Heller (2021) generalized quasi-ordinal knowledge spaces to polytomous items. Inspired by this paper, we propose CD-polytomous knowledge space and its polytomous surmise system. A Galois connection is established between the collection $K$ of all polytomous knowledge structures and the collection $F_1$ of particular polytomous attribute functions. The closed elements of the Galois connection are CD-polytomous knowledge spaces in $K$ and polytomous surmise functions in $F_1$, respectively. With the help of these, this paper provides a characterization of the polytomous knowledge structure corresponding to the polytomous surmise function that is weakly factorial. Based on the finite sets of items and response values, these results generalize the previous approaches for polytomous knowledge spaces.