On Galois connections between polytomous knowledge structures and polytomous attributions

Polytomous knowledge structure theory (abbr. polytomous KST) was introduced by Stefanutti et al. (2020) and further results on polytomous KST were obtained by Heller (2021). As the interesting work, this paper discusses Galois connections in polytomous KST. In this paper, two derivations between polytomous knowledge structures and polytomous attributions are presented. In addition, this paper gives an explicit characterization to introduce the completeness of polytomous attributions and defines the concept of a complete polytomous knowledge structure by the property that such a polytomous knowledge structure is derived from a complete polytomous attribution. This paper establishes a Galois connection between the collection $K$ of all polytomous knowledge structures and the collection $F$ of all polytomous attributions, where the closed elements are respectively in $K$ the complete polytomous knowledge structures, and in $F$ the complete polytomous attributions. Furthermore, this Galois connection induces a one-to-one correspondence between the two sets of closed elements. Moreover, this Galois connection can also induce a Galois connection between the collection of all granular polytomous knowledge structures and the collection of all granular polytomous attributions.