Constructing polytomous knowledge structures from L-fuzzy S-approximation operators

Rough set theory primarily focuses on the characteristics of upper and lower approximations of specific sets, rather than their overall structure. Knowledge space theory can provide a new perspective on rough sets. In recent years, this theory has introduced polytomous knowledge structures, which have emerged as a significant and innovative concept in the field. This paper embeds L-fuzzy sets in S-approximation spaces and establishes a connection between polytomous knowledge structures and L-fuzzy S-approximation operators. We generate polytomous knowledge structures using these operators, present their corresponding properties, and show that a polytomous knowledge space and a polytomous closure space can be fully characterized by an upper and lower L-fuzzy S-approximation, respectively. In particular, we discuss four special L-fuzzy S-approximation operators and relate them to existing fuzzy skill maps. Subsequently, we further investigate the construction of two specific dichotomous knowledge structures, called backward-graded and forward-graded, using one of these four L-fuzzy S-approximation operators. We want to offer a new viewpoint for analyzing the structures of L-fuzzy S-approximation spaces through the lens of knowledge space theory.