Knowledge structure theory was established by Doignon and Falmagne (1985), which conceives items as being dichotomous. It categorizes the responses into the dichotomy of “mastering” vs. “not mastering” an item (or, “solving” vs. “not solving”, respectively). Recently, Stefanutti et al. (2020) and Heller (2021) extended the binary case to a (complete) lattice (L,≤) where L is the set of response values and ≤ is a partial order on L, which can be identified with different types, or degrees of mastering. Moreover, they provided some characterizations of the resulting polytomous knowledge structures that are based on the domain Q consisting of items and the (complete) lattice (L,≤) consisting of response values. Such findings open the field to a systematic generalization of knowledge structure theory to the polytomous case.

In knowledge structure theory, a number of one-to-one correspondences were established between particular collections of mathematical structures. Specially, Doignon and Falmagne (1999) extended the one-to-one correspondence between quasi ordinal spaces and quasi orders obtained by Birkhoff (1937) and established a one-to-one correspondence between granular knowledge spaces on the domain Q and surmise functions on Q (also see Theorem 5.2.5 in Falmagne & Doignon, 2011). Recently, as an adaptation to the polytomous framework of the Doignon–Falmagne’s result, Stefanutti et al. (2020) gave the following theorem.

Theorem 11 inStefanutti et al. (2020). For the set Q of items and the complete lattice (L,≤) of response values, let K⊔ and M⊔ denote the collection of all granular polytomous spaces on (Q,L) and the collection of all polytomous surmise functions on (Q,L), respectively. Then there is a one-to-one correspondence between K⊔ and M⊔. It is defined, for K∈K⊔ and μ∈M⊔ by the formula:

C is an atom at q in K ⟺ C is a clause for q in μ,

Unfortunately, the above theorem does not hold true. In fact, there exists a counterexample to show that there is not any one-to-one correspondence between the collection of all granular polytomous spaces on (Q,L) and the collection of all polytomous surmise functions on (Q,L) for some set Q of items and some complete lattice (L,≤) of response values. Here, the counterexample stems from the fact that the function m:K⊔⟶M⊔ described in the proof of the above theorem need not be surjective. As an interesting work, our discussion spreads from polytomous surmise functions. It should be noticed that these discussions do not affect the rest of Stefanutti et al. (2020).

In this paper, we modify the concept of polytomous surmise functions to introduce polytomous surmising functions that are stronger than polytomous surmise functions. Moreover, an example is given to show that a polytomous surmise function need not be a polytomous surmising function. Let Q be a set of items and (L,≤) a complete lattice of response values. We prove that there exists a one-to-one correspondence f between the collection of all granular polytomous spaces on (Q,L) and the collection of all polytomous surmising functions on (Q,L), where polytomous surmising functions cannot be replaced with polytomous surmise functions. This result gives a correction of Theorem 11 in Stefanutti et al. (2020). In addition, as an application of the correspondence f, we show that the pair (f,f−1) of mappings forms a Galois connection where all granular polytomous spaces and all polytomous surmising functions are closed elements of this Galois connection.