Knowledge space theory (KST) is a mathematical theory first proposed by Doignon and Falmagne, 1985, Doignon and Falmagne, 1988. It determines students’ cognitive level in different knowledge by analyzing their answers to a series of relevant questions at different levels. Based on theories such as pedagogy and psychology, KST has established a set of mathematical theories to reflect the laws of education, which provides an effective scientific method for educational evaluation. KST is also a theory to test students’ knowledge states and construct their knowledge structures.

In the early stage of KST, knowledge spaces were at the focus of their development. It is pretty reasonable to assume that two students interacting for a while will eventually merge their initial state of knowledge into a single state which is the union of the two. Doignon and Falmagne (1985) generalized Birkhoff’s Theorem (Birkhoff, 1937) and showed that (granular) knowledge spaces are in a one-to-one correspondence with surmise systems. AND/OR graphs of artificial intelligence (Nilsson, 1971) can be used to intuitively understand the relationship between items and foundations in a surmise system. Koppen (1998) introduced additional conditions on surmise functions and investigated their consequences for the corresponding spaces (especially for well-graded knowledge spaces). Ge and Lin (2021) further studied attribution functions (esp. surmise functions) from the perspective of discriminativeness and resolubility. Shen et al. (2021) introduced the completeness (completion) of attribution functions that were in a one-to-one correspondence to knowledge spaces.

Classic KST assumes that an individual’s answer to an item is true or false. This dichotomous representation has limitations in the evaluation of knowledge and learning. To address this limitation, Schrepp (1997) generalized the main concepts of KST to problems with more than two answers, using linear ordered sets to evaluate the quality of solutions. Bartl and Belohlavek (2011) discussed the theory of knowledge spaces with graded knowledge states. Stefanutti et al. (2020) introduced a new formulation of the polytomous KST, assuming that the response levels form a complete lattice. Heller (2021) extended the quasi-ordinal knowledge space to the polytomous case and incorporated the methods of Schrepp and Stefanutti et al. into his generalization structure. Wang et al. (2022a) explored a specific polytomous knowledge structure: CD-polytomous knowledge space, and it can be regarded as the extension of knowledge space in the polytomous case. Other notes on polytomous generalization can be referred to Wang et al. (2022b).

Heller (2021) introduced the extended polytomous knowledge structure, the element of which is the extended polytomous knowledge state. Each extended polytomous state represents all polytomous items that a subject can solve. This paper presents an extended surmise system, and it takes advantage of the characteristics of the extended polytomous knowledge structure and then defines the clause for each polytomous item. Naturally, it is a generalization of Heller’s concepts of (pre-)precedence relations, allowing more than one possible learning foundation for a polytomous item. This paper also shows the relationship between the extended surmise systems and the CD-polytomous knowledge spaces. In addition, Wang et al. (2022a) proposed the concept of a polytomous surmise system. Although the structures generated by the two surmise systems may be the same, the extended surmise system in this paper is different from that of Wang et al. who consider the polytomous surmise system from the perspective of polytomous knowledge structure. The article is structured as follows: Section 2 provides relevant concepts of KST. Section 3 proposes the extended surmise system and Galois connection, and Section 4 gives some notes on the polytomous surmise systems and the (pre-)precedence relations. A final discussion is in Section 5.