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A Unified Framework for Polytomous Knowledge Structures via Fuzzy Attribute Functions
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DOI：10.1016/j.fss.2026.110019.png)
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Abstract
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We present a unified way to derive polytomous knowledge states from fuzzy attributes on arbitrary finite posets. A fuzzy compatibility condition alone ensures that the induced map yields single-valued states, whereas, under a suitable regularity assumption, the combination of both fuzzy compatibility and fuzzy completeness guarantees that a fuzzy attribute structure generates a full polytomous knowledge structure. For the conjunctive model, an order-theoretic regularity condition guarantees closure under existing pointwise meets; for the disjunctive model, we identify when pointwise joins are preserved. We also construct maximal structures and provide representation results that connect responses to fuzzy profiles. The framework subsumes binary Knowledge Space Theory (KST), fuzzy skill functions on lattices, and non-fuzzy poset attribute maps, offering a unified perspective on graded responses. A three-item example from combinatorics and probability illustrates how the framework represents partial mastery and misconception-driven responses.
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