Knowledge Structure Theory (KST), originally developed for dichotomous items by Doignon and Falmagne, 1985, Doignon and Falmagne, 1999, has since been extended to polytomous items. The foundational framework of polytomous knowledge structure theory began with Schrepp (1997); since then, it has evolved significantly through the contributions of numerous researchers, including Heller (2021) and Stefanutti, Anselmi, de Chiusole and Spoto (2020), and more recently, Stefanutti, Spoto, Anselmi, and de Chiusole (2023).

These researchers have formalized the concepts of polytomous knowledge states and structures, expanded the fundamental notions of surmise relations and surmise functions from classical KST, and established specific Galois connections within polytomous structures. A key motivation for these extensions is to address the limitations of classical dichotomous KST, which assumes a strict binary division between mastery and non-mastery of items. By incorporating polytomous response models, researchers have introduced a more flexible and refined framework that enables knowledge states to distinguish multiple intermediate levels. This extension is particularly useful in educational assessments, where learners’ progress often follows a gradual trajectory rather than an abrupt transition between discrete states.

More recent developments have placed increasing emphasis on expanding the theoretical and methodological foundations of polytomous knowledge structures. One primary focus has been the generalization of surmise functions to the polytomous setting (Wang et al., 2023, Wang et al., 2023a), defining them on item-response level pairs and their clauses to better model different solution pathways across response levels. Additionally, significant theoretical advancements have been made in the establishment of comprehensive Galois connections (Ge, 2022, Wang et al., 2023a), which establish one-to-one correspondences between various surmise functions and certain polytomous knowledge structures. Researchers have also investigated well-gradedness properties in polytomous settings (Sun et al., 2023, Wang and Li, 2024), ensuring smooth transitions between adjacent polytomous knowledge states to better reflect realistic learning processes. Another important development involves data-driven methodologies, including k-median clustering algorithms and probabilistic models designed to enhance the practical applicability of polytomous KST (de Chiusole et al., 2020, Stefanutti, de Chiusole, Anselmi et al., 2020). These approaches enable the extraction of meaningful polytomous knowledge structures directly from response data, reducing reliance on predefined expert assumptions and making the framework more adaptable to diverse educational and psychological assessment contexts.

In addition to these advancements, competence-based knowledge structure theory (CbKST; e.g., Doignon, 1994, Düntsch and Gediga, 1995, Falmagne et al., 2013, Gediga and Düntsch, 2002, Heller et al., 2015, Korossy, 1997, Korossy, 1999) has gained considerable traction, particularly through the use of skill maps or skill functions. These maps define the relationship between problems and the required skills, providing a structured framework for both knowledge representation and competence assessment. Different theoretical perspectives guide how these relationships are interpreted. In some models, problem-solving is assumed to require the combined mastery of all relevant skills, reflecting a cumulative learning process. In contrast, alternative approaches allow for more flexibility, recognizing that possessing any one of the necessary skills may be sufficient to solve a given problem. The choice between these perspectives influences how knowledge states are structured and how individual learning trajectories are assessed. As knowledge assessments increasingly adopt polytomous response formats, it becomes essential to extend these models beyond the dichotomous setting, enabling a more nuanced understanding of competence progression.

Building on this foundation, Stefanutti et al. (2023) expanded the concept of skills to encompass more general attributes, introducing polytomous attribute maps constrained by a set of conditions. They developed attribute structures, item–response functions, and constructed polytomous knowledge structures over more general partially ordered sets. As they noted, their attribute map is conjunctive in nature, meaning all assigned attributes must work together to reach an observable response level. However, in practical polytomous contexts, the conjunctive model is not always applicable, as attributes may sometimes function independently, each capable of enabling the item to reach a response level on its own.

For instance, consider solving quadratic equations of the form mx2+nx+p=0, where m, n, and p are real numbers. A student who has mastered the factorization method — considered here as an attribute, denoted by a — may be able to solve simpler equations with integer or rational roots, thereby achieving only a partial score. Furthermore, a student who has mastered the quadratic formula (attribute b) or the completing-the-square method (attribute c) can typically solve any form of quadratic equation, thereby achieving full marks.

Hence, there is a need to investigate a different type of attribute map suited for such models. This paper introduces a new form of attribute map, constrained by four conditions to ensure its applicability. Consequently, the construction of attribute structures and item–response functions must follow these constraints, leading to an appropriate polytomous knowledge structure.

The paper is organized as follows: Section 2 introduces preliminary knowledge, Section 3 presents the main contributions, including the new attribute map and the corresponding item–response function, and Section 4 offers a mathematical example to illustrate the model’s applicability. Section 5 compares the proposed model with the existing conjunctive approach. Finally, Section 6 provides a concluding discussion.