In the theory of knowledge structures (KST), individual ability is represented by a so-called ‘knowledge state’, which is the set of all the problems in a given domain Q an (hypothetical) individual is capable of solving (Doignon & Falmagne, 1999; Falmagne & Doignon, 2011). The collection of all the knowledge states, named the ‘knowledge structure’, is a multidimensional model of individual ability in the domain under consideration. It is natural to consider that a knowledge structure is one-dimensional when its knowledge states form a chain of sets, that is, when it is a ‘chain space’. More generally any definition of a dimension parameter generates a classification of knowledge structures according to their dimension, which can then serve as a rough assessment of their complexity. In many cases, the definition also conveys an embedding of a knowledge structure into some geometric space. A link with measurement theory appears, generating potentially valuable insights. These questions drive our investigation into the notion of dimension from a theoretical perspective.

From the applicative viewpoint, KST is gradually expanding beyond its original educational domain into new areas of application. There are successful applications of its concepts and models to psychological, cognitive, and neuropsychological assessment (Brancaccio et al., 2023, Chiusole et al., 2024, Spoto et al., 2010, Stefanutti, 2019, Stefanutti et al., 2021). Traditional and modern psychometric approaches commonly use a notion of dimension to capture explanatory latent factors. Also in KST, a competence-based approach exists where “explanatory latent factors” are dichotomous or polytomous skills or attributes (Doignon, 1994, Heller, Augustin, et al., 2013, Heller, Ünlü, and Albert, 2013, Stefanutti et al., 2023). However, the relationship between skills and dimensions is not clear. By providing a precise notion of dimension, we aim at better relating KST to other established psychometric theories. This, in turn, could facilitate exchange and understanding of KST from external perspectives, allowing others to appreciate its advantages over competing approaches.

What could be a sound notion of ‘dimension’ for knowledge structures? Let us briefly sketch the four options that we consider in this paper.

Ordinal dimension. Knowledge structures are families of subsets and, as such, they are partially ordered by set inclusion. For partial orders, there exist various definitions of dimension in the literature (see for instance Dushnik and Miller, 1941, Kelly and Trotter, 1982, Roberts, 1985, Trotter, 1992). The ‘dimension’ of a partial order, is the most fundamental concept. Doignon and Falmagne (1988) define the ordinal dimension of a knowledge structure as the dimension of the poset formed by its states taken with the inclusion relation.

Knowledge structures are models which find applications in the assessment of individual knowledge. Given any two knowledge states K and L, the relation K⊆L has a clear interpretation from an educational or cognitive perspective: if K is the knowledge state of a student s, and L is the knowledge state of a student t, then t is capable of solving all of the problems that s can solve, plus eventually some other problems. While the ordinal dimension has a precise, formal definition, its psychological interpretation may not be that natural or immediate. If a knowledge structure has ordinal dimension n, then there are n linear orders ≤1, ≤2, …, ≤n on the collection of knowledge states whose intersection is the restriction of the inclusion relation ⊆ to this collection. If n≥2 there exist, for each i=1, 2, …, n, two states K and L which are incomparable with respect to set inclusion and such that K≤iL. The interpretation of ≤i is not immediate in this case.

Spatial dimension. We introduce here a second type of dimension, which may have a more natural interpretation in an educational context, but which applies only to ‘knowledge spaces’—knowledge structures in which any union of states is again a state. By its definition, the spatial dimension of the knowledge space (Q,K) is the least number of chain spaces in the lattice of all knowledge spaces on Q whose least upper bound is K. It is akin to the ‘convex dimension’ of convex geometries from Edelman and Jamison (1985). For recent work on various dimension concepts for convex geometries, the reader is referred to Knauer and Trotter (2024).

The successive states of a chain space represent the progress of a student acquiring step after step the mastery of the whole domain under study. The spatial dimension of a knowledge space is well in line with the closure under union of the collection of knowledge states. It captures the least number of student lines of progress needed to produce by the union operator each of the knowledge states in the space.

The intersection- and the union-bidimensions. Two more notions of dimension of a knowledge structure rely on the theory of biorders (Doignon, Ducamp, & Falmagne, 1984), particularly the ‘bidimension’ of a binary relation. The intersection-bidimension (resp. union-bidimension) of a relation is the smallest number of biorders having the relation as their intersection (resp. union). Doignon and Falmagne (1988) define the intersection-bidimension of a knowledge structure as the intersection-bidimension of its membership relation—and similarly for the union-bidimension.

The membership relation of a knowledge structure is a biorder exactly if the knowledge states form a chain. More generally, it has intersection-bidimension n exactly if it is an intersection of n membership relations of chain spaces. Thus a conjunctive model combining in some precise way n chain spaces explains the membership relation. The interpretation of the union-bidimension is similar with a disjunctive model.

The four dimension parameters we just introduced all rely on a precise way of combining chain spaces. Because they rely on different combinations, it should not be a surprise that they often (but not always) produce distinct values for the dimension of a same structure. The bulk of our work is to provide proofs for the inequalities between two of the dimension parameters which are always valid, and to produce counter-examples for the other inequalities.

The article is organized as follows. Section 2 provides the necessary background on KST, on the dimension of a poset, and on the two bidimensions of a binary relation. Section 3 gives the definitions of the ordinal and spatial dimensions, as well as some preliminary results. The next section is about the two variants of the bidimension of a knowledge structure and their relationships to the two other dimension parameters. Section 5 introduces a special type of knowledge structures, named “terse”. In terse knowledge spaces the ordinal dimension happens to be equal to the spatial dimension. Section 6 is about the dimension of learning spaces. The final remarks in Section 7 summarize our results. Four diagrams show the valid inequalities under four different assumptions on the knowledge structures. The section also lists some open problems.