A new type of polytomous surmise system

Doignon and Falmagne (1985) introduced a surmise system, which generalized the precedence relation, allowing multiple possible learning paths for an item. Heller (2021) took into account precedence relations on an extended set of (virtual) items and further generalized quasi-ordinal knowledge spaces to polytomous items. Wang et al. (2022) proposed CD-polytomous knowledge space and provided its corresponding polytomous surmise system. Following these developments and drawing upon the so-called extended polytomous knowledge structure, this paper presents two concepts: weak polytomous structure and extended surmise system. Via setting up a Galois connection, a one-to-one correspondence is established between the collection of all extended surmise functions and the collection of certain weak polytomous structures. This paper also comprehensively discusses the relationships among the precedence relations, the polytomous surmise systems, and the extended surmise systems.