Knowledge Structure Theory (KST), initially developed by Doignon and Falmagne, 1985, Doignon and Falmagne, 1999, has undergone significant advancements by other researchers (Falmagne et al., 2013, Falmagne and Doignon, 2011). Originally focused on dichotomous items, KST has been extended to incorporate items with multiple response alternatives.

Schrepp (1997) generalized KST to handle items with multiple response alternatives, while Stefanutti et al. (2020) established a robust mathematical foundation for extending KST to items with more than two ordered response categories using linearly ordered sets and complete lattices. Heller (2021) further extended quasi-ordinal knowledge spaces to polytomous items by incorporating partially ordered response values into lattices. Stefanutti et al. (2023) developed an axiomatic theory of attribute maps, extending skill maps to polytomous knowledge structures and establishing a deterministic relationship between attributes and observable item responses. These advancements have significantly advanced the understanding and application of KST, offering diverse mathematical frameworks and perspectives for assessing knowledge across various contexts. This expansion has also sparked substantial research dedicated to exploring polytomous KST.

In the realm of polytomous knowledge structures, Wang et al. (2023) and Ge (2022) established Galois connections between polytomous knowledge structures and polytomous attribute functions. However, due to differing interpretations, their findings yielded distinct closed elements. Specifically, Wang et al. focused on component-directed polytomous knowledge spaces, contrasting with Stefanutti et al. (2020)’s more rigorous concept of polytomous knowledge spaces by adopting a more flexible criterion for the join operation. Moreover, Sun, Li, He, et al. (2023) contributed to the understanding of polytomous knowledge structures by extending the notion of well-gradedness to the polytomous case.

The integration of fuzzy mathematics and polytomous KST has brought about significant advancements. Bartl and Belohlavek (2011) proposed using fuzzy sets to represent knowledge states, with membership degrees indicating individual mastery levels. Zhou et al. (2022) focused on constructing personalized learning paths through polytomous structures based on item and fuzzy skill relationships. Sun, Li, Lin, and He (2023) developed specialized algorithms for constructing polytomous knowledge structures, exploring the connection between fuzzy skills and polytomous items.

Turning our attention to well-graded families in the field of knowledge structures, the concept of well-graded knowledge structures was initially introduced and refined by Doignon and Falmagne (1997) in the finite case. This exploration of well-gradedness stemmed from a stochastic theory developed by Falmagne and Doignon (1988), which has implications for the development of efficient algorithms for knowledge assessment (Eppstein et al., 2009). Cosyn and Uzun (2009) further demonstrated the equivalence between learning spaces and well-graded knowledge spaces. Matayoshi (2017) also examined various properties of well-graded union-closed families that exclude the empty set. In addition to their theoretical properties, well-graded families hold practical significance and serve as the foundation for artificial intelligence systems such as the ALEKS system (Falmagne et al., 2006, Falmagne and Doignon, 2011).

In the field of polytomous KST, there have been initial advancements in the study of well-graded structures. Heller (2021) introduced a metric δp, on a finite graded lattice, which enables the measurement of distance δ between polytomous knowledge states. Expanding on this metric, Sun, Li, He, et al. (2023) utilized it to define and explore the concept of well-gradedness in the polytomous case. It is worth noting that Ovchinnikov (2002) proposed an alternative concept of well-gradedness, which exhibits conciseness and certain geometric properties when applied to finite linear orders.

This paper adopts a rigorous approach to the subject matter and builds upon the works of Sun, Li, He, et al. (2023) and Ovchinnikov (2002), aiming to provide a comprehensive understanding of well-graded families in the context of polytomous KST. By thoroughly analyzing the definition of well-gradedness proposed by Sun et al. we offer an extensive overview of this concept. Additionally, we propose an equivalent characterization of well-graded families based on the insights provided by Ovchinnikov. Furthermore, we explore various manifestations of well-gradedness in polytomous knowledge structures, including the examination of strong discriminative properties, the representation of well-gradedness in polytomous knowledge spaces, and the well-gradedness property of discriminative factorial polytomous knowledge structures. Through the research, our aim is to contribute to the advancement and deeper understanding of this critical aspect within the field of Knowledge Structure Theory.

The remaining sections of the paper are organized as follows: In Section 2, we introduce key concepts in KST and discuss the extension of KST to incorporate polytomous items. Section 3 presents our proposed characterization of well-gradedness in a general non-empty family. Moving forward, in Section 4, we delve into the examination of various manifestations of well-gradedness specifically in polytomous knowledge structures. Finally, in Section 5, we summarize the main findings and contributions of our research.