On the polytomous generalization of knowledge space theory

Journal of Mathematical Psychology 94 (2020) 102306

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On the polytomous generalization of knowledge space theory ∗

Luca Stefanutti a , , Pasquale Anselmi a , Debora de Chiusole a , Andrea Spoto b a b

Department of Philosophy, Sociology, Pedagogy and Applied Psychology, University of Padua, Italy Department of General Psychology, University of Padua, Italy

article

info

Article history: Received 13 April 2017 Received in revised form 31 October 2019 Accepted 8 November 2019 Available online xxxx Keywords: Knowledge space theory Polytomous items Polytomous structure Beliefs Quasi orders Surmise systems

a b s t r a c t One of the core assumptions of knowledge space theory (KST) is that the answer of a subject to an item can be dichotomously classified as correct or incorrect. Schrepp (1997) provided a very first attempt to generalize the main KST concepts to items with more than two response alternatives, but his work has not had a strong impact on the subsequent research on KST. The aim of the present article is to introduce a new formulation of the polytomous KST, starting from the work of Schrepp and broadening it to a wider extent. Schrepp’s generalization is revisited, and the fundamental closure conditions are reformulated and decomposed into a necessary and sufficient set of four independent properties of polytomous knowledge structures. Among them, two special properties emerge in the polytomous case that in the dichotomous one are neither testable nor immediately visible, since necessarily true. These properties allow for a straight generalization of Birkhoff’s Theorem with respect to quasi-ordinal knowledge spaces, and Doignon and Falmagne’s Theorem for knowledge spaces. Such findings open the field to a systematic generalization of many KST concepts to the polytomous case. © 2019 Elsevier Inc. All rights reserved.

1. Introduction One of the core assumptions of knowledge space theory (KST; Doignon & Falmagne, 1985, 1999) is that the answer of a subject to an item can be dichotomously classified as correct or incorrect. This assumption appears to be very well suited for the classical field of application of KST, namely the adaptive assessment of knowledge, where the knowledge state of an individual can be described as the set of items that he or she masters. The generalization of the classical concepts to more than two alternatives could represent the stepping stone for the application of KST to a number of different fields (e.g., psychological assessment, attitudes and opinions, partial credit items, response times, etc.) in which the dichotomous case appears to be too restrictive. Nevertheless, such generalization has not been carried out yet to a large extent. So far, two different attempts for generalizing KST to nondichotomus item responses have been conducted. The former was proposed twenty years ago by Schrepp (1997), the latter was proposed more recently by Bartl and Belohlavek (2011). The present work is mainly based on the article by Schrepp, although it has also connections to the ‘‘fuzzy’’ approach by Bartl and Belohlavek. In both extensions, the items in a set Q are ‘‘evaluated’’ through levels in a set L, indicating the quality of the solution for ∗ Corresponding author. E-mail address: [email protected] (L. Stefanutti). https://doi.org/10.1016/j.jmp.2019.102306 0022-2496/© 2019 Elsevier Inc. All rights reserved.

q in an ordinal way. In ordinary (dichotomous) KST, the collection L would contain exactly two ‘‘levels’’. Polytomous extensions, instead, admit that L contains more than two alternatives. A knowledge state is then defined as a function K : Q → L that assigns levels to items. The two approaches differ by the specific assumptions on the set L. In the development proposed by Schrepp, L is a linearly ordered set. In the approach by Bartl and Belohlavek it is a complete residuated lattice, that is a complete lattice which is also a commutative monoid. The direction taken here is somehow half the way between the two, since L is assumed to be any complete lattice. If L is infinite and it is not a lattice, then least upper bounds need not exist, and this may constitute a problem. In this specific aspect, the development followed here is more restrictive than that by Schrepp. It is however not so restrictive as the one proposed by Bartl and Belohlavek. In fact, it is not assumed here that L is a commutative monoid. Such an assumption would require the existence of a kind of ‘‘concatenation’’ or ‘‘aggregation’’ operator ⊗ among levels in L such that, given any two levels a, b ∈ L, a ⊗ b are also in L. For the type of empirical systems that can be encountered in the social and behavioral sciences, such requirement seems to be too strict. For instance, it is not trivial to establish how two levels of a rating scale such as ‘‘partial disagreement’’ and ‘‘partial agreement’’ can concatenate to produce a third level of that rating scale. As stated above, Schrepp proposes a reformulation of the main elements of KST with the aim of having, for each item q in a given domain Q , more than two levels, taken from a set L, each of which indicates, in an ordinal way, the quality of the solution for

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q. Therefore, a knowledge state is regarded as a way of assigning levels in L to items in Q , and it is thus defined as a mapping K : Q → L. Starting from this fundamental definition, Schrepp provides a generalization of the concepts of knowledge structure, knowledge space, and quasi-ordinal knowledge space. Finally, he shows that knowledge structures need to fulfill stronger closure properties in the polytomous case than in the dichotomous one, in order to be in a one-to-one correspondence with the surmise relations and functions. Schrepp’s generalization to polytomous items has not had a strong impact on the subsequent research on KST. This may be due to some particular reasons. First, as mentioned above, the main field of application of KST (i.e., the assessment of knowledge) can be fruitfully represented by a dichotomous theory. Second, in 1997 KST was a relatively young theory and some of its main concepts were not completely defined. The main theoretical reference, that is the book by Doignon and Falmagne (1999), was not published yet. Thus, researchers were not ready to an extension of a brand new theory. In any case, it is quite natural that a dichotomous assessment theory is, sooner or later, extended to polytomous items. This was at least the case of item response theory (Lord, 1980; Rasch, 1960–1980), that was initially developed for the dichotomous case, but was lately extended to the polytomous one (e.g., Andrich, 1978). In KST this passage appears today to be particularly urgent and necessary since, in the last years, an increasing interest has been devoted to the application of the theory in fields like psychological assessment (e.g., Bottesi, Spoto, Freeston, Sanavio, & Vidotto, 2015; Spoto, Bottesi, Sanavio, & Vidotto, 2013) and social sciences (Martin & Wiley, 2000; Wiley & Martin, 1999) which often require polytomous items. The aim of the present article is to provide new theoretical results to the generalization of KST to the polytomous case, starting from the work of Schrepp and broadening it to a wider extent. We revisit Schrepp’s generalization and characterize the fundamental conditions, decomposing them into a necessary and sufficient set of independent properties of ‘‘polytomous’’ knowledge structures. Among them, two special properties emerge in the polytomous case that in the dichotomous case are not visible, since necessarily true. These properties allow for a straight generalization of Birkhoff’s Theorem (Birkhoff, 1937) with respect to quasi-ordinal knowledge spaces, and Doignon and Falmagne’s Theorem (Doignon & Falmagne, 1985) for knowledge spaces. Such findings open the field to a systematic generalization of many KST concepts to the polytomous case. The article is structured as follows: Section 2 provides a recap of the main key-concepts of KST; Section 3 introduces the generalization of KST to polytomous items; Section 4 provides an example of application on empirical data; A final discussion is provided in Section 5. 2. Basic concepts in knowledge space theory In KST, the set of items that can be asked about a specific topic is the knowledge domain Q . A knowledge state K ⊆ Q is the set of items of the domain that an individual masters. The set of all the admissible knowledge states is the knowledge structure K, and it contains at least ∅ and Q . Whenever a knowledge structure is closed under set-union, it is called a knowledge space. Whenever a knowledge space is also closed under set-intersection, it is a quasi-ordinal knowledge space. The theorem by Birkhoff (1937) establishes a one-to-one correspondence between the set of all the quasi-ordinal knowledge spaces defined on Q and the set of all the surmise relations (i.e., quasi-order relations) on Q . Moreover, Doignon and Falmagne (1985) established a one-toone correspondence between the set of all the so-called ‘‘granular knowledge spaces’’, and all the surmise functions defined on Q .

Q

Definition 1. A mapping σ : Q → 2(2 ) , assigning to each q ∈ Q a family of subsets of Q , is named a surmise function if it satisfies the following four conditions: (i) (ii) (iii) (iv)

if if if if

q ∈ Q , then σ (q) ̸ = ∅; C ∈ σ (q), then q ∈ C ; q′ ∈ C ∈ σ (q), then C ′ ⊆ C for some C ′ ∈ σ (q′ ); C , C ′ ∈ σ (q) and C ′ ⊆ C , then C = C ′ .

In such case the pair (Q , σ ) is named a surmise system and each set C ∈ σ (q) is said a clause for q. What differentiates between a surmise relation and a surmise function is that the former admits only one clause for each item. A very fundamental concept in KST is that of a notion, which is a maximal set of Q such that any two of its items belong to exactly the same states. All the items contained a same notion are said to be equally informative (Doignon & Falmagne, 1999) since they all convey the same information. Of course, the collection of all the notions is a partition of the domain Q of the structure. Whenever all the notions of a knowledge structure are singletons, the structure is said to be discriminative. All the concepts introduced so far have been defined and formalized for dichotomous items. In the following sections it will be shown how all these basic elements can be generalized to the polytomous case allowing for a brand new extension of KST. 3. Spaces for polytomous items Let Q be a set of items and (L, ⩽) be a complete lattice, that is an ordered set in which every subset S ⊆ L has both an infimum (greatest lower bound) and a supremum (least upper bound). For instance, L could be any closed interval [a, b] of the real numbers or any finite linearly ordered set. However, in general it is not required that L is a numeric set. Typical examples in the social and behavioral sciences are rating scales, where levels are linearly ordered and usually have a linguistic description (like, e.g., ‘‘total disagreement–partial disagreement–partial agreement–total agreement’’). Examples of what L cannot be: the set R of the real numbers because R lacks both an infimum and a supremum; any open or half open interval. Drawing upon the work by Schrepp (1997), a state is regarded as a way of assigning levels in L to items in Q , and it is thus defined as a mapping K : Q → L. The term state is intentionally left rather general and unspecific: Depending on the application, further qualifications can be attached to it (e.g., knowledge state in the standard KST, belief state in the approach of Wiley & Martin, 1999, etc). Following Davey and Priestley (2002), the collection of all the mappings from Q to L is denoted LQ . The definition of a state as a mapping from items to levels induces a natural ordering, sometimes termed pointwise order (see, e.g., Davey & Priestley, 2002), on the collection LQ : Given two mappings K1 , K2 ∈ L Q , K1 ⊑ K2 ⇐⇒ ∀q ∈ Q : K1 (q) ⩽ K2 (q). It is not difficult to see that ⊑ is indeed a partial order (i.e., reflexive, transitive, and antisymmetric relation) on the set LQ of mappings. Indeed, LQ is itself a complete lattice, since infima and suprema exist for all of its subsets in virtue of the existence of infima and suprema for all subsets of L. In the sequel, the term dominance relation will be used for referring to this partial order so that, for instance, if K1 ⊑ K2 then we say that K1 is dominated by K2 (or that K2 dominates K1 ). When both L and Q are finite sets, any state ⟨ K : Q → ⟩L can be represented as a |Q |-tuple of the form v1 , v2 , . . . , v|Q | , with positions representing items in Q and such that, for i = 1, 2, . . . , |Q |, vi = K (qi ). In such cases, to simplify the notation,

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

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a |Q |-tuple v1 , v2 , . . . , v|Q | may be sometimes represented as v1 v2 · · · v|Q | .

⟨

⟩

A polytomous structure on the two sets Q and L is a triple (Q , L, K) where K is a nonempty collection of states (e.g., all the states existing in a certain population, at a given moment in time). The structure is finite if it has a finite number of states. We notice that, depending on the properties of the two sets Q and L, the polytomous structure K may be either finite or infinite. If both Q and L are finite then K is obviously finite. In the practice of item development, it is often required that an order relation ≾ exists for the items in a given set Q , that is, the items can at least be ordered. This is the case in many self report questionnaires for psychological assessment in which individual’s level of agreement is asked with respect to a number of questions. Given any two items p, q ∈ Q , we can write p ≾ q to mean that a respondent who disagrees with p, also disagrees with q (or that an agreement with p is a prerequisite for an agreement with q). Consider the following two items taken from the Rathus Assertiveness Schedule (Rathus, 1973): 1. I can easily express my opinions. 2. If, during a conference, a well-known and respected speaker would state something that I think is wrong, I would express my opinion. Given these two items it could be hypothesized that an individual who disagrees with Item 1 would also disagree with Item 2. Formally, the relation ‘‘being a prerequisite of’’ is well represented by a reflexive and transitive relation on Q , that is a quasi-order. If this type of order on the items exists, then not all ways of assigning levels in L to items in Q may be compatible – in a well-specified sense – with the hypothesized order. Compatibility of states with the order ≾ on the items, in turn, depends on how the quasi-order ≾ on the items in Q is ‘‘linked’’ to the linear order ⩽ on the levels in L. Definition 2. Let ≾ be a quasi-order (i.e., reflexive and transitive binary) relation on the set Q of items. A state K : Q → L is compatible with ≾ if p≾q

H⇒

K (q) ⩽ K (p)

(1)

for all p, q ∈ Q . The collection K of all the states that are compatible with ≾ is named the quasi-ordinal polytomous space induced by ≾. Call (1) the monotonic rule of agreement. This definition differs from the one provided by Schrepp (1997) only for the direction of the inequality that appears in the righthand side term of the implication in (1). Schrepp takes it in the opposite direction (i.e., K (p) ⩽ K (q)). This change is motivated by the different application context: ‘‘quality of the solution in knowledge assessment’’ in Schrepp’s work, and ‘‘degree of agreement with a set of items’’ (and thus, assessment of beliefs, attitudes, etc.) in this article. It should also be noticed that the direction considered in Eq. (1) is coherent with the one usually assumed in the dichotomous case (where p ≾ q whenever q ∈ X implies p ∈ X for all X ∈ X , with X being a family of subsets of Q ). In line with this shift of perspective, the term belief state (Martin & Wiley, 2000; Wiley & Martin, 1999) will be sometimes used to refer to the states in K. Example 1. Let L = {0, 1, 2} be a set of three levels linearly ordered by ≤, and Q = {a, b, c } be a set of three items for which the quasi-order ≾ is defined such that a ≾ b and a ≾ c. The collection of all the belief states that are compatible with ≾ is displayed in the Hasse diagram of Fig. 1. Call K1 this collection. In the figure each state K is represented by a triple xyz of values with x = K (a), y = K (b) and z = K (c). Thus, for instance, the triple 212 represents the state K with K (a) = 2, K (b) = 1 and K (c) = 2.

Fig. 1. Hasse diagram of the quasi-ordinal polytomous space (Q , L, K1 ), with Q = {a, b, c }, and L = {0, 1, 2}. K1 is the collection of all the states that, according to the monotonic agreement rule, are compatible with the partial order ≾ such that a ≾ b and a ≾ c.

A very basic concept in the dichotomous KST is that of equally informative items (Doignon & Falmagne, 1999). In the polytomous framework, this notion of equally informative items can be generalized as follows: Given the polytomous knowledge structure (Q , L, K), and any two items p, q ∈ Q , p ∼ q ⇐⇒ ∀K ∈ K : K (p) = K (q). The equivalence relation ∼ induces equivalence classes on the items q ∈ Q , that are denoted q∗ = {p ∈ Q : p ∼ q}, and named beliefs (in the standard dichotomous KST they are named notions). Again, the distinction is justified by the change of context. The set of all the beliefs is a partition on the set Q : Q ∗ = {q∗ : q ∈ Q }. A structure K is said to be discriminative if every belief q∗ ∈ Q ∗ is a singleton. 3.1. Properties of quasi-ordinal polytomous spaces There is a quite well-known result that holds true in the dichotomous case (i.e., when L = {0, 1}), which identifies the combinatorial properties characterizing the collection of states that are compatible with a given quasi-order ≾ on the set Q . The result is a theorem by Birkhoff (1937), that establishes a one-to-one correspondence between the collection of all quasiorders on Q and the family of all collections of subsets of Q that are closed under both union and intersection. The question is whether Birkhoff’s Theorem generalizes to the polytomous case (i.e., |L| > 2). In the dichotomous case, union and intersection are operations whose results are respectively the supremum and the infimum of their operands. This characteristic is preserved in the polytomous case by the following two operations. Definition 3. Given a polytomous structure⨆ (Q , L, K), and any subfamily G ⊆ K , define the two mappings G : Q → L and d G : Q → L such that, for all q ∈ Q ,

(⨆ )

G (q) = sup{K (q) : K ∈ G },

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L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

and (l ) G (q) = inf{K (q) : K ∈ G }, where, for X ⊆ L, sup X and inf X are the supremum and the infimum of X in L. By convention, the mapping ∅L : Q → L such that ∅L (q) = 0 for all q ∈ Q ⨆ is the supremum of the empty subfamily of ⨆ F , that is ∅L = ∅. Then K is closed under the supremum d if G ∈ K for all G ⊆ K, and it is closed under the infimum if G ∈ K for all G ⊆ K. The polytomous structures that are closed under the supremum are named the polytomous spaces. When G = ⨆ {K1 , K2 } with K1 , K2 ∈ K arbitrary, d the notation K1 ⊔ K2 is used for {K1 , K2 }, and K1 ⊓ K2 stands for {K1 , K2 }. A simple inductive argument shows that a finite structure K is closed under the supremum if and only if K1 ⊔ K2 ∈ K for all K1 , K2 ∈ K, and it is closed under the infimum if and only if K1 ⊓ K2 ∈ K for all K1 , K2 ∈ K. Considering again the structure K1 displayed in Fig. 1, it can be easily verified that it turns out to be closed under both the supremum and the infimum, supporting the intuition that, once ∪ and ∩ have been replaced by ⊔ and ⊓, Birkhoff’s Theorem generalizes to the polytomous case in the obvious way. But, as it was shown by Schrepp (1997), it is not so. Structures which are both ⊔-closed and ⊓-closed need not be in a one-to-one correspondence with the family of all quasi-orders on Q . Hence, the quasi-ordinal polytomous spaces must be characterized by some more restrictive condition. Such a condition was identified by Schrepp, and named strict closure.1 Definition 4. A structure (Q , L, K) is said strictly closed if, for all mappings F ∈ LQ , the condition

∀p, q ∈ Q , (∀K ∈ K : K (p) ⩽ K (q)) H⇒ F (p) ⩽ F (q)

(2)

implies F ∈ K. Let R and K be, respectively, the collection of all the quasiorders on Q , and that of all the strictly closed polytomous spaces on Q and L. A bijective correspondence between the two sets can be established. To this aim, we define two mappings k and r. Let k : R → K be such that, for R ∈ R and any F ∈ LQ , F ∈ k(R) ⇐⇒ ∀(p, q) ∈ R : F (q) ⩽ F (p). On the other side, let r : K → R be such that, for K ∈ K and p, q ∈ Q , (p, q) ∈ r(K) ⇐⇒ ∀K ∈ K : K (q) ⩽ K (p). The following theorem generalizes Theorem 3 by Schrepp (1997) in two directions. Firstly, it sets up a bijective correspondence between R and K. In particular, it shows that both k : R → K and r : K → R are onto their respective codomains. Secondly, it extends the result to arbitrary complete lattices L. Theorem 1. The two mappings k : K → R and r : R → K are mutually inverse bijective correspondences between the family K of the strictly closed spaces on (Q , L), and the collection R of all the quasi orders on Q . Proof. Let ‘◦’ denote function composition. Condition (2) is equivalent to

∀(p, q) ∈ Q 2 : (p, q) ∈ r(K) H⇒ F (q) ⩽ F (p), which further simplifies to ∀(p, q) ∈ r(K) : F (q) ⩽ F (p), which holds true if and only if F ∈ (k ◦ r)(K). Thus, from strict closure 1 It should be noticed that the definition of strict closure provided by Schrepp (1997) differs from the present one. As observed by Jean-Paul Doignon (personal communication), in Schrepp’s definition parentheses are misplaced.

of K it follows that ∀F ∈ LQ : F ∈ (k ◦ r)(K) H⇒ F ∈ K, or more concisely, (k ◦ r)(K) ⊆ K. On the other side, from F ∈ / (k ◦ r)(K) it follows that there exists (p, q) ∈ Q 2 such that (p, q) ∈ r(K) and F (q) ̸ ⩽ F (p), which implies F ∈ / K. Thus F ∈ K implies F ∈ (k◦r)(K). Namely K ⊆ (k◦r)(K). This shows that (k◦r)(K) = K for all K ∈ K. Let p, q ∈ Q and R ∈ R. If (p, q) ∈ R then K (q) ⩽ K (p) for all K ∈ k(R), which in turn implies (p, q) ∈ (r ◦ k)(R). Thus R ⊆ (r ◦ k)(R). Assume, on the contrary, that (p, q) ∈ / R and define K ′ : Q → L such that, for x ∈ Q , K ′ (x) =

{

sup L

if (x, q) ∈ R

inf L

if (x, q) ∈ / R.

′

To show K ∈ k(R), suppose, by contradiction, there is a pair (x, y) ∈ R with K ′ (y) ̸ ⩽ K ′ (x). This holds if and only if K ′ (x) = inf L and K ′ (y) = sup L. In turn, K ′ (x) = inf(L) iff (x, q) ∈ / R whereas K ′ (y) = sup L iff (y, q) ∈ R. It follows from the transitivity of R that (x, y) ∈ / R, which is a contradiction. Hence K ′ (x) ⩽ K ′ (y) for all (x, y) ∈ R and thus K ′ ∈ k(R). Then (p, q) ∈ / R entails K ′ (p) = inf L. Since K ′ (q) = sup L, we see that there is K ′ ∈ k(R) with K ′ (q) ̸ ⩽ K ′ (p). Therefore (p, q) ∈ / (r ◦ k)(R) and, hence (r ◦ k)(R) ⊆ R. This shows that (r ◦ k)(R) = R for all R ∈ R. □ All along the article we will refer to strictly closed structures as the ‘‘quasi-ordinal polytomous spaces’’. The quasi-order ≾ that corresponds to a given strictly closed structure K on Q and L is obtained by setting p ≾ q if and only if K (q) ⩽ K (p) holds true for all K ∈ K (K (p) ⩽ K (q) in the Schrepp’s formulation). A quasiordinal polytomous space that turns out to be discriminative is named an ordinal polytomous space. Notice in passing that any constant mapping K : Q → L (i.e., such that K (q) = l ∈ L for all q ∈ Q ) belongs to any quasi-ordinal polytomous space. This fact generalizes the condition according to which every (dichotomous) space contains the empty set and the full domain Q . There is an apparent discontinuity between the dichotomous and the polytomous frameworks on how quasi-ordinal spaces are recognized, distinguished from other structures and, thus, characterized. In the dichotomous case, a quasi-ordinal space K can be recognized by only looking at subsets, or families of subsets that are internal elements of K (i.e., just check that all possible unions and intersections of subsets in the structure are also in it). In the polytomous case, for testing the strict closure condition it is not sufficient to consider the internal elements of a structure only. A verification of the condition across all the |L||Q | different mappings F : Q → L in LQ cannot be avoided. The ⊔-closure and the ⊓-closure still play a role in characterizing quasi-ordinal polytomous spaces, in the sense that they are necessary, though insufficient conditions. For Q and L finite, this was already observed by Schrepp (1997, Lemma 3). In the infinite case (remembering that L is assumed to be a complete lattice), the following result holds. Theorem 2. Let Q be any set of items, L be any complete lattice, and ≾ a quasi-order on Q . Then, the collection K of all mappings K : Q⨆→ L satisfying Condition (1) of Definition 2 is closed under d both and . Proof. Closure under : suppose ⨆ that p, q ∈ Q are such that p ≾ q, and let F ⊆ K. Set K = F and, by contradiction, assume K (q) > K (p). Then for any state F in F we get F (q) ≤ F (p) ≤ K (p) < K (q). We have a contradiction with the fact that K (q) has to be the leastdupper bound in L of all F (q), for F in F . For the closure under same line of reasoning is followed, after ⨆ , the d having replaced by , F u by the collection of the lower bounds F ℓ. □

⨆

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

⨆

d

Closure under or under may not hold if L is not a complete lattice. To give an example, suppose L = [0, 1) ∪ (2, 3]. Then L is not a complete lattice because, for instance, the open interval [0, 1) misses a supremum. For Q = {p, q}, suppose p ≾ q and let K be the collection of all mappings K : {p, q} → L satisfying Condition (1). Then K is essentially the set {(x, y) ∈ L2 : y ≤ x}. The set F = {(x, y) ∈ [0, 1)2 : y ≤ x} is a subset of K. Its set of upper bounds is F u = (2, 3]2 . It is clear that there is no least element (x, y) in this set, hence F has no least upper bound. As a consequence, K cannot be closed under the supremum. Closure under the infimum and closure under the supremum can be viewed as parts of the strict closure condition, and the question is whether strict closure can be decomposed into a set of independent properties that are jointly necessary and sufficient. Two candidates are ⊔-closure and ⊓-closure, but which are the missing ones? An answer to this question is given in Sections 3.3 and 3.4. 3.2. Atoms and bases In the dichotomous theory of knowledge structures, among all the states containing a given item q ∈ Q , those that are minimal with respect to set inclusion are called atoms at q. The collection of all the states in K that are atoms at some q ∈ Q is named the basis. If K is a quasi-ordinal space and B is its basis then (B, ⊆) is isomorphic to (Q , ≾), meaning that each item in Q is represented by a unique atom in B and that the order on the atoms, with respect to ⊆, is the same as that on the items, with respect to ≾. This is how ∩- and ∪-closed structures represent quasi-orders in the dichotomous case. The notions of atoms and bases need to be appropriately extended to the polytomous theory. Such extension relies upon certain fundamental concepts in lattice theory (see, e.g., Davey & Priestley, 2002). First of all, since (L, ⩽) is a complete lattice, it is bounded below, that is it contains a least level inf L (i.e., such that inf L ⩽ l for all l ∈ L) which, by convention, will be represented by the symbol ‘0’. Secondly, the notion of ‘‘join-irreducible’’ level of the complete lattice L is also needed. A single level x ∈ L is said to be join-irreducible (or, simply, irreducible) if, for all subsets G ⊆ L, sup G = x implies x ∈ G. By convention, the supremum of an empty set of levels is 0 (sup ∅ = 0), therefore the infimum level of L is not irreducible. An infinite lattice need not possess join-irreducible levels. An example is the unit interval [0, 1]: any x ∈ [0, 1] is such that x = sup[0, x). A sufficient condition for the existence of the irreducible levels is the so-called descending chain condition (DCC), stating that for every descending chain x1 ≥ x2 ≥ · · · ≥ xn ≥ · · · of levels in L, there is a natural number n such that, for all k ≥ n, xk+1 = xk , that is the chain stabilizes. Of course DCC always holds if L is finite. Definition 5. Let (Q , L, K) be a polytomous structure and let L+ denote the subset of the all the irreducible levels of L. For l ∈ L and q ∈ Q , a state K ∈ K is an l-atom at q if the following three conditions are all satisfied: (1) l ∈ L+ ; (2) K (q) = l; (3) K ′ ⊑ K and K ′ (q) = l imply K ′ = K , for all K ′ ∈ K. Then K is said to be an atom at q if it is an l-atom at q for some l ∈ L+ . Moreover, it is called an l-atom if it is an l-atom at some q ∈ Q. Thus, K is an l-atom at q if it is minimal w.r.t. ⊑ among all those states assigning level l to q. There are no l-atoms for non-irreducible levels. Moreover, in a structure (Q , L, K) with

5

Table 1 Atoms in the structure K1 . Item

a b c

Atoms at levels 1

2

100 110 101

200 220 202

Q infinite, atoms may not exist. This may happen in both the dichotomous and the polytomous frameworks. For the dichotomous case see Example 1.27 in Doignon and Falmagne (1999). It is worth noticing that, if L is a finite linear order (like, e.g., many rating scales) then every level in L but 0 is irreducible so that L+ = L \ {0}. Example 2. The structure K1 of Example 1 turns out to have six different atoms, which are listed in Table 1. Items in Q are listed in rows, whereas levels in L+ are listed in columns. Each entry of the table contains an l-atom at q for the row q and the column l. Having extended the notion of an atom to the polytomous case, we now proceed by generalizing the notion of a basis. Definition 6. ⨆ Given a polytomous structure (Q , L, K), a mapping K ∈ K is said -irreducible (or irreducible to supremum) if, for all ⨆ subfamilies F ⊆ K , F = K implies K ∈ F . The subfamily B ⨆ of all the -irreducible mappings in K is named the basis of the polytomous structure. The following theorem is a generalization of Theorem 1.20 in Doignon and Falmagne ⨆ (1999) to the polytomous case. Since the collection of all the -irreducible levels is necessarily unique, the proof follows at once from the definition of a basis. Theorem 3. A polytomous structure (Q , L, K) admits at most one basis. In the dichotomous theory of knowledge structures, the collection of all the atoms of a knowledge structure is the basis of the structure. The next two theorems show that this is still valid in the polytomous framework. Theorem 4. A state ⨆ in a polytomous structure (Q , L, K) is an atom if and only if it is -irreducible. Proof. Sufficiency: If K is not an atom then for every q ∈ Q there must exist a state Jq ∈ K with Jq (q) = K (q) and Jq ⊏ K . Defining the collection F = {Jq : q ∈ Q } of all such states we have ⨆ F = K and K ∈ / F . Necessity: For l ∈ L+⨆ , assume K ∈ K is an latom at q⨆ ∈ Q and let F ⊆ K be such ⨆that F = K . In particular, one has ( F )(q) = K (q) = l. But ( F )(q) = sup{F (q) : F ∈ F }. Since l is an irreducible level, from {F (q) : F ∈ F } ⊆ L and sup{F (q) : F ∈ F } = l it follows that l ∈ {F (q) : F ∈ F }, meaning that there is F ∈ F with ⨆ F (q) = l. Furthermore, it must be that F ⊑ K because K = F . However F ⊏ K must be false, for otherwise K would not be an l-atom. Hence K ∈ F immediately follows. □ The following result now follows at once. Corollary 1. The basis of a polytomous structure is the collection of all of its atoms. In the dichotomous theory, some infinite structures have no basis. Moreover, even when the basis exists, there may be no atoms at some of the items (see, e.g. Examples 1.21 and 1.27 in Doignon and Falmagne (1999)). The so-called granularity of the

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structure assures the existence of a basis and the existence of atoms at each of the items. The notion of granularity is needed in the polytomous theory too, though in a more general guise. Definition 7. A polytomous structure (Q , L, K) is said to be granular if for any state K ∈ K, any item q ∈ Q , and any level l ∈ L \ {0}, K (q) = l implies that there is an l-atom at q that is dominated by K . In the dichotomous KST, any finite knowledge structure is granular. This may not hold true in the polytomous case, as the following example points out. Example 3. Let Q = {p, q} and L = {a, b, c , d} be, respectively, the set of items and the set of levels. Let moreover ⩽ be a partial order on L such that a ⩽ b, a ⩽ c , b ⩽ d, c ⩽ d, and ≾ be a quasiorder on Q such that p ≾ q. It is easily verified that (L, ⩽) is a complete lattice. In particular, d = sup{b, c } is the greatest level, and a = inf{b, c } is the least level. Using the notation K (p)K (q) for representing the states, the quasi-ordinal polytomous space corresponding to ≾ is K = {aa, ba, bb, ca, cc , da, db, dc , dd}.

Then we notice that the state da is not an atom, because d is a join-reducible level, and a is the least level, which is reducible by definition. Moreover, there is no d-atom in K dominated by da, simply because there exists no d-atom at all in K. Thus K is not granular. An infinite polytomous structure may fail to be granular for the same reasons as those of an infinite dichotomous structure, plus some additional ones. In fact, if L+ is a strict subset of L \ {0} (a situation that never arises in the dichotomous case, or in the finite case), then there exist no l-atoms for anyone of the levels l ∈ L \ (L+ ∪ {0}). As a consequence, any state K ∈ K assigning to an item q ∈ Q any one of such levels, falsifies the granularity of the structure. There is one further property of the basis that holds in the dichotomous case and may not hold in the polytomous one. In the dichotomous theory, if a knowledge space possesses a basis B, then it equals the so-called ‘‘span’’ of B, which is the collection of all possible unions of subfamilies of B (Definition 1.19 in Doignon & Falmagne, 1999). This property of the basis may no longer hold with polytomous structures, and the nature of the set L of levels is crucial in this respect. To clarify this, the notion of ‘‘span’’ need be extended to the polytomous structures. Definition 8. The span of a family F ⊆ LQ of mappings is the family F ′ ⊆ LQ containing any mapping which is the supremum ⨆ G of some subfamily G ⊑ F . In such case we say that F spans F ′. Granularity⨆is a sufficient condition under which a polytomous space (i.e., a -closed polytomous structure) is spanned by its basis, as established by the following theorem. Theorem 5. If (Q , L, K) is a granular polytomous space then it is the span of its basis. Proof. Let B be the basis of K and S be the span of B. For K ∈ K and q ∈ Q define the supremum

ˆ Kq =

⨆

{B ∈ B : B ⊑ K , B(q) = K (q)}.

If K (q) = 0 then we either have (i) B(q) = 0 for all B ⊑ K and hence ˆ Kq (q) =⨆0, or (ii) there is no B ⊑ K with B(q) = 0 and thus ˆ Kq (q) = ∅ = 0. In both cases, ˆ Kq (q) = K (q). If instead K (q) ̸ = 0 then, by the granularity of K, there exists B ∈ B

with B ⊑ K and B(q) = K (q). Thus, ˆ Kq (q) = K (q). Moreover, if ˆ r(⨆ ∈ Q \ {)q} is any other item, then ˆ Kr (q) ⨆ ⪯ K (q) = Kq (q). Thus, ˆ ˆ K (q) = K (q). It follows that K = K . Since this r ∈Q r q∈Q q conclusion holds for any K ⨆ ∈ K, we have that K ⊆ S . On the other side, let F ⨆ ⊆ B. Then F ∈ K immediately follows from B ⊆ K and the -closure of K itself. Thus S ⊆ K. This shows that K is the span of B. □ The next two examples show, respectively, one case where the basis spans the whole structure and one case where the span of the basis is a proper subset of the whole structure. Example 4. The infinite set L = {1/n : n ∈ N} ∪ {0} is a complete lattice. Let (Q , L, K) be a polytomous structure with Q = {a, b}, and K = {K ∈ L2 : K (a) ≥ K (b)}. Consider any level u ∈ L \ {0}, any subset X ⊆ L and suppose that sup X ≤ u (i.e., u is an upper bound of X ) and u ∈ / X . This holds if and only if x < u for all x ∈ X . Since there is n ∈ N with u = 1/n, there is u′ ∈ L such that u′ = 1/(1 + n) and {l ∈ L : u′ < x < u} = ∅. Thus x < u implies x ≤ u′ for all x ∈ X . Therefore sup X < u. This shows that u must be irreducible and that, indeed, L+ = L \ {0}. Given any l ∈ L+ , the set of l-atoms of K is B(l) = {(l, 0), (l, l)}, and the basis is the infinite set B=

⋃ (1) B

n∈N

n

.

Given any (x, y) ∈ K, we can always find (x, 0) in B(x) and (y, y) ∈ B(y) with (x, 0) ⊔ (y, y) = (x, y), because x ≥ y. Thus B spans K. In the infinite case, if granularity does not hold, then there may be states K ∈ K and items q ∈ Q such that K (q) ∈ L \ (L+ ∪ {0}). In such cases there would be no ⨆ K (q)-atoms in the basis and, thus, no subfamilies F ⊆ B with ( F )(q) = K (q). Therefore the basis would only span a strict subset of K. The following example shows one such case. Example 5. Let L = [0, 1/3] ∪ [2/3, 1]. An arbitrary point x ∈ L is irreducible if and only if x ∈ / X implies sup(X ) ̸= x for all X ⊆ L. Let X ⊆ {x ∈ L : x < 2/3}. Observing that x < 2/3 implies x ≤ 1/3 for all x ∈ L, it follows that X ⊆ {x ∈ L : x ≤ 1/3}. Thus sup X < 2/3 and therefore 2/3 is an irreducible level. On the other side take any y ∈ L with y ̸ = 2/3. Of course, if y = 0 then it is reducible. If y ∈ [0, 1/3] then sup[0, y) = y and hence y is not reducible. An analogous conclusion is drawn for y ∈ (2/3, 1]. Therefore L+ = {2/3}. Letting Q = {a, b} and K = {K ∈ L2 : K (a) ≥ K (b)}, we see that the basis is B = {(2/3, 0), (2/3, 2/3)}. The span of B is {(0, 0), (2/3, 0), (2/3, 2/3)} ⊂ K. Granularity has rather strong consequences. In the first place, it implies that L is a total order, otherwise, any two incomparable levels i and j in L would have a join-reducible supremum l (leading to the impossibility of having l-atoms). Furthermore, every level i ∈ L must have a ‘‘predecessor’’ (i.e., a level j ∈ L such that j < i and there is no l ∈ L with j < l < i), otherwise i would be the supremum of a subset not containing it and hence it would be join-reducible. Definition 9. Let (Q , L, K) be a polytomous structure with basis B. For any l ∈ L+ , the collection of all the l-atoms in K is named the l-basis of K and is denoted B(l). Moreover, for q ∈ Q , denote with B[q] the collection of all elements in B that are atoms at q: B[q] = {B ∈ B : B is an atom at q}.

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

It is clear that the basis of K is simply the union of all the l-bases: B=

⋃

B(l).

l∈L+

For instance, the space K1 of Example 1 has two l-bases, one for each of the two levels in L+ = {1, 2}: B1 (1) = {100, 110, 101} B1 (2) = {200, 220, 202}.

Thus, the basis of K1 turns out to be B1 = {100, 110, 101, 200, 220, 202}.

To conclude the section, the following theorem establishes a fundamental property of the l-atoms of a granular quasi-ordinal polytomous space. Theorem 6. In a granular quasi-ordinal polytomous space (Q , L, K), there is exactly one l-atom at q for each item q ∈ Q and each level l ∈ L \ {0}. Proof. For q ∈ Q and l ∈ L \ {0}, let Kql = {K ∈ K : K (q) = l} be the collection of all states in K assigning level l to item q. This collection is not empty because, for instance, the constant mapping K : Q → L such that K (q) = l for all q ∈ Q satisfies the monotonic rule of agreement in (1) of Definition 2 and hence it d is in K. Furthermore, since Kql ⊆ K and K is -closed, it holds d that Kql ∈ K. Thus, by the granularity of K, for every K ∈ Kql there is an atom B ∈ K such that B ⊑ K and B(q) = K (q). From d B ∈ Kql it follows that Kql ⊑ B. Strict dominance cannot hold, d for otherwise B would not be an l-atom. Hence Kql = B. Its uniqueness follows from uniqueness of the infimum. □ 3.3. Isomorphic l-bases and level-free structures It can be verified in Example 1 that there is an orderisomorphism h between the two l-bases B1 (1) and B1 (2) (where the order relations are restrictions of ⊑) that is obtained by setting h(100) = 200, h(110) = 220, and h(101) = 202. This isomorphism has the property that B ∈ B1 (1) and its image h(B) are both atoms at the same item. Thus, 100 and h(100) = 200 are both atoms at a, 110 and h(110) are both atoms at b, 101 and h(101) = 202 are both atoms at c. Hence we can say that the 1-atoms in the basis B1 (1) are represented by the 2-atoms in the basis B1 (2) in the sense that (i) each of the 1-atoms is uniquely represented by a 2-atom, and that (ii) the order on the 1-atoms is the same as that on the 2-atoms. In other words, the basis B(l), as a partially ordered set, turns out to be independent of the particular level l ∈ L+ of the ordinal scale (L, ⩽) that one may consider. Definition 10. Given a polytomous structure (Q , L, K) having basis B and any two levels i, j ∈ L+ , the two l-bases B(i) and B(j) of K are said to be isomorphic to one another if there is a bijection h : B(i) → B(j) such that: (1) A ⊑ B ⇐⇒ h(A) ⊑ h(B) for all A, B ∈ B(i), (2) h(B[q] (i)) = B[q] (j), where for F ⊆ B, h(F ) = {h(F ) : F ∈ F }. The structure K is said to be level-free if all of its l-bases are pairwise isomorphic. Condition (1) of Definition 10 states that the bijection h is, indeed, an order isomorphism of B(i) onto B(j). Condition (2) of Definition 10 states that, for every atom B ∈ B(i), B and h(B) must be atoms for the same item(s).

7

Besides being isomorphic to one another, the two l-bases of K1 in Example 1 are also isomorphic to (Q , ≾), and this is indeed the most important and critical property. The isomorphism f : B1 (1) → Q is trivially obtained by setting f (100) = a, f (110) = b and f (101) = c. To summarize, we observe that the following properties hold true for the structure K1 : It is both ⊓- and ⊔-closed and it is levelfree. The following examples show that these three properties are independent of one another. Example 6. Consider the following three polytomous structures on Q = {a, b, c } and L = {0, 1, 2}: K2 = {000, 100, 111, 200, 211, 220, 221, 222}, K3 = {000, 100, 110, 111, 200, 222}, K4 = {000, 100, 110, 200, 211, 220, 221, 222}.

Structure K2 provides an example of a ⊔-closed structure which is neither ⊓-closed nor level-free. For example, it is not ⊓-closed because the element-wise infimum of 211 and 220 is 210 which is not in K2 . Moreover, K2 is not level free, because the 1-basis is {100, 111}, the 2-basis is {200, 220, 222}, and hence there is no bijection between the two. Concerning K3 , it is ⊓-closed but neither ⊔-closed nor level-free. In fact, 211 = 200 ⊔ 111 is not in K3 . Finally, K4 is level-free, but it is neither ⊓- nor ⊔-closed. These examples make thus clear that level-freedom, ⊓-closure and ⊔-closure are independent properties of a structure. Theorem 7. Let (Q , L, K) be a quasi-ordinal polytomous space with basis B. For any level l ∈ L+ and any two items q, r ∈ Q , let Bq , Br ∈ B(l) be l-atoms at q and r respectively. Then Bq ⊑ Br if and only if K (r) ⩽ K (q) for all K ∈ K. Proof. Sufficiency: Let the condition ∀K ∈ K : K (r) ⩽ K (q) hold true. Then we have Br (r) ⩽ Br (q). But, since Bq and Br are l-atoms, Bq (q) = Br (r) = l must hold and thus Bq (q) = Br (r) ⩽ Br (q). Suppose Bq (s) > Br (s) holds for some s ∈ Q \ {q}. Then, by the ⊓-closure of K, Bq ⊓ Br is a state in K, and (Bq ⊓ Br )(q) = Bq (q) = l. But then, since Bq ⊓ Br ⊑ Bq and Bq ⊓ Br ̸ = Bq , Bq would not be an l-atom at q. Thus Bq (s) ⩽ Br (s) must hold for all s ∈ Q and hence Bq ⊑ Br must be concluded. Necessity: Suppose K (q) < K (r) for some K ∈ K and let K (q) = i and K (r) = j so that i < j. Let moreover Bqi = inf{J ∈ K : J(q) = i} and Brj = inf{J ∈ K : J(r) = j}. Then Bqi ⊔ Brj ⊑ K . Thus we have: (Bqi ⊔ Brj )(q) ⩽ K (q) = i and (Bqi ⊔ Brj )(r) ⩽ K (r) = j. But Bqi (q) = i and Brj (r) = j by construction and hence, indeed, (Bqi ⊔ Brj )(q) = i and (Bqi ⊔ Brj )(r) = j. This entails the following sequence of inequalities: Brj (q) ⩽ Bqi (q) = i < j = Brj (r). Let Bqj be the j-atom at q, so that Bqj (q) = j = Brj (r). Then Bqj (q) = Brj (r) > Brj (q) implies Bqj ̸ ⊑ Brj . Since, indeed, Bqj is j-atom at q and Brj is j-atom at r this concludes the proof. □ It should be observed that the previous result may not hold for polytomous spaces that are not quasi-ordinal. In a quasi-ordinal space, every item has a unique l-atom for every l ∈ L+ . This assumption does no longer hold in a polytomous space which is not quasi-ordinal. In such a space an item may have more than a single l-atom and the condition ‘‘K (r) ⩽ K (q) for all K ∈ K’’ may happen to be either unnecessary or insufficient. Consider for instance the polytomous space K for Q = {p, q, r } and L = {0, 1, 2} having basis B = {100, 200, 010, 020, 101, 202, 011, 022}. In this space, the inequality K (r) ⩽ K (p) is false for some K ∈ K (e.g., 011 or 022). However 100 is a 1-atom for p, 101 is a 1-atom for r and 100 ⊑ 101. Thus the condition at issue is not necessary. A second example is the following one: with L = {0, 1, 2} and Q = {p, q, r , s}, let the basis be B = {1100, 0101, 0111, 2200, 0202, 0222}. Then, K (r) ⩽ K (q) holds

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L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

for all states K in the polytomous knowledge space obtained by closing this basis. However, given the two 1-atoms Bq = 1100 and Br = 0111 we have Bq ̸ ⊑ Br , showing that the condition is not sufficient. We recall that the equivalence classes induced by the symmetric part of the quasi-order ≾ are named beliefs (see Section 3). Corollary 2. In a quasi-ordinal polytomous space (Q , L, K), at every level l ∈ L+ there exists a bijection gl : Q ∗ → B(l) mapping beliefs in Q ∗ to atoms in B(l). Proof. Let l ∈ L+ . By Theorem 6, for each item q ∈ Q there exists a unique l-atom Bq . For q, r ∈ Q , let Bq and Br be l-atoms at q and r respectively. By the definition of a belief we have q∗ = r ∗ if and only if both K (q) ⩽ K (r) and K (r) ⩽ K (q) are true for all K ∈ K. By Theorem 7 these are the necessary and sufficient conditions for both Bq ⊑ Br and Br ⊑ Bq and, hence, for Bq = Br . Thus, defining gl (q∗ ) = Bq we see that q∗ = r ∗ if and only if gl (q∗ ) = gl (r ∗ ). Thus gl is injective. Moreover, since every B ∈ B(l) is l-atom at some q ∈ Q , gl is also onto and thus it is a bijection. □ Theorem 8. Every quasi-ordinal polytomous space K on the two sets Q and L is level-free. Proof. By Corollary 2, for every level l ∈ L+ there is a bijection gl : Q ∗ → B(l). Hence, given i, j ∈ L+ , also the function composition hij = gj ◦ gi−1 is a bijection mapping B(i) onto B(j). Of course, this bijection satisfies Condition (2) of Definition 10 by construction. Thus, we only have to show that A ⊑ B iff hij (A) ⊑ h(B) for all A, B ∈ B(i). But this now easily follows from Theorem 7. □ Thus, level-freedom is another necessary condition of the strictly closed structures. It is indeed an interesting and desirable property: If each l-basis B(l) is regarded as a representation of the order on the items, then this representation does not change as one moves from one level to another of the rating scale (L, ⩽). Using different words, what this property states is that, if the structure K is aimed at representing a certain order on the set Q of items, then the represented order must be the same across all levels in L+ . 3.4. Independent levels In view of characterizing the quasi-ordinal spaces, the set of the three properties ⊓-closure, ⊔-closure and level-freedom is still insufficient, as the following example points out. Example 7. The structures K5 = {000, 100, 110, 111, 211, 221, 222}, K6 = {000, 100, 200, 210, 220, 221, 222},

are two other structures satisfying all three properties: They are ⊓-closed, ⊔-closed, and level-free. Their bases are B5 (1) = {100, 110, 111}, B5 (2) = {211, 221, 222}, B6 (1) = {100, 210, 221}, and B6 (2) = {200, 220, 222}. The two structures K5 and K6 contain less states than the structure K1 of Example 1 (see Fig. 1). They are indeed both subsets of K1 . Thus some of the states satisfying the compatibility condition provided in Definition 2 are missing in either K5 (like, e.g., 200) or K6 (e.g. 110). At a closer look, we can see that structure K5 allows certain types of inferences that are not allowed by K1 . For instance, let K be any state in K5 . If we know that K assigns a 2 to item a, then we must conclude that it assigns at least a 1 to both items b and c. Hence the additional rule K (a) > 1 H⇒ ∀q ∈ Q : K (q) ≥ 1

holds true in K5 , but it is false in K1 since, for instance, 200 is a state in K1 . This is why K5 is, in a sense, more restrictive than K1 . On the other side, K6 satisfies the additional conditions K (b) > 0 H⇒ K (a) = 2 and K (c) > 0 H⇒ K (b) = 2. Of course, these additional conditions are not required to be true by the compatibility condition in Definition 2. Thus the question is: Among all the structures that are level-free and closed under the infimum and the supremum, which are the least restrictive ones? Definition 11. Given a polytomous structure (Q , L, K) having basis B, a level l ∈ L+ is said to be independent in K if B(q) > 0 implies B(q) = l for all B ∈ B(l) and q ∈ Q . If every level l ∈ L+ is independent in K then we say that K has independent levels. It can be seen that K1 has independent levels, whereas K5 and K6 do not. As a property, independence of levels is independent of the other three properties. The structure K7 = {000, 100, 110, 111, 002, 022, 222}

has l-bases B7 (1) = {100, 110, 111} and B7 (2) = {002, 022, 222} and, thus, it has independent levels. However it is neither ⊓- nor ⊔-closed. For instance, 110 ⊓ 022 = 010 which is not in K7 and 100 ⊔ 002 = 102 is not in K7 . Finally, the structure is not even level-free, because there is no bijection between the two bases that satisfies both Conditions (1) and (2) of Definition 10. Theorem 9. Every quasi-ordinal polytomous space K on the two sets Q and L has independent levels. Proof. Define the function β : Q → 2Q such that, for q ∈ Q ,

β (q) = {r ∈ Q : K (q) ⩽ K (r) for all K ∈ K}. Given any q ∈ Q and any arbitrary level l ∈ L+ , define the mapping Aq : Q → L such that, given r ∈ Q , l

if r ∈ β (q),

0

if r ∈ Q \ β (q).

{ Aq (r) =

We show that Aq is an l-atom at q in K. Take any arbitrary r , s ∈ Q such that K (r) ⩽ K (s) for all K ∈ K. If K (q) ⩽ K (r) for all K ∈ K then r , s ∈ β (q) and hence Aq (r) = Aq (s) = l, which is consistent with the strictly closure condition. On the other hand, if K (r) < K (q) for some K ∈ K and K (q) ⩽ K (s) for all K ∈ K then r ∈ / β (q) and s ∈ β (q), entailing that Aq (r) = 0 < l = Aq (s), which is also consistent with the strictly closure condition. Finally, if K (s) < K (q) for some K ∈ K then s ∈ / β (q). Moreover, since K (r) ⩽ K (s) for all K ∈ K and K (s) < K (q) for some K ∈ K, it follows that K (r) < K (q) for some K ∈ K. Hence r ∈ / β (q). Thus Aq (r) = Aq (s) = 0, which is still consistent with the strictly closure condition. We conclude that Aq ∈ K. To see that Aq is an l-atom at q, suppose there is B ⊏ Aq with B(q) = l. Then, it follows at once from the definition of the mapping Aq that there must be r ∈ β (q) with B(r) < l = B(q), which violates the strict closure condition (because K (q) ⩽ K (r) for all K ∈ K), entailing B ∈ / K. Thus Aq must be an l-atom at q. Since there is an l-atom of the form Aq for every level l ∈ L+ and every item q ∈ Q and every item has exactly one l-atom, it must be concluded that atoms can only have the form of Aq in a strictly closed structure, entailing that in a strictly closed structure all levels are independent. □

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

Thus, independence of the levels is another necessary condition in strictly closed structures. Theorem 10 collects all the necessary and sufficient properties of the strictly closed structures. Lemma 1. Let (Q , L, K) be a quasi-ordinal polytomous space with basis B. Given two items q, r ∈ Q and a level l ∈ L+ , let Bq , Br ∈ B(l) be l-atoms at q and r respectively. Then Bq (r) = l if and only if Br ⊑ Bq . Proof. Necessity: Since Br is l-atom at r, from Bq (r) = l it follows that (Bq ⊓ Br )(r) = l. But Bq ⊓ Br ⊑ Br and the fact that Br is l-atom at r imply that Bq ⊓ Br = Br and, hence Br ⊑ Bq . Sufficiency: From Br ⊑ Bq we have l = Br (r) ⩽ Bq (r) ∈ {0, l}. Hence Bq (r) = l. □ Theorem 10. A granular polytomous structure (Q , L, K) is a (granular) quasi-ordinal polytomous space if and only if all the following four conditions hold true: (i) (ii) (iii) (iv)

K K K K

is level-free, has independent levels, d is ⨆-closed, is -closed.

Proof. Necessity of properties (i), (ii), (iii) and (iv) is clear. Then we have to show that (i)–(iv) jointly imply that K is strictly closed or, equivalently, by Theorem 1, that the equality (k ◦ r)(K) = K holds true. Thus, suppose all conditions (i)–(iv) hold in K. Hence we know that both K and (k ◦ r)(K) are closed under the infimum and under the supremum. Therefore, since by Theorem 5 any granular polytomous space is the span of its basis, we only need ˜ = (k ◦ r)(K) and to proof that they have identical bases. Define K ˜ . Since, for l ∈ L+ let B be the basis of K, and B˜ be the basis of K arbitrary, each notion in Q ∗ has a unique l-atom in B(l), we can form a bijection gl : Q ∗ → B(l) assigning to each notion q∗ ∈ Q ∗ its l-atom gl (q∗ ) ∈ B(l). Then by Theorem 7 and the definition of the function r, for every p, q ∈ Q , gl (p∗ ) ⊑ gl (q∗ ) ⇐⇒ K (q) ⩽ K (p) for all K ∈ K

⇐⇒ pr(K)q. Hence we see that gl is an order-isomorphism of (Q , r(K)) onto ˜ is strictly closed, the same arguments (B(l), ⊆). Moreover, since K used for K lead to the conclusion that, for l ∈ L+ , there is an ˜ ) = r(K)) onto order-isomorphism hl : Q ∗ → B˜(l) of (Q , r(K (B˜(l), ⊆). Hence for every l ∈ L+ the composition fl := hl ◦ gl−1 is an order-isomorphism of (B(l), ⊆) onto (B˜(l), ⊆). We now show that, indeed, fl is the identity. For q∗ ∈ Q ∗ , let Bq = gl (q∗ ) and B˜ q = hl (q∗ ). By applying Lemma 1 and by the fact that fl is an isomorphism, we obtain the following chain of equivalences: Bq (p) = l iff Bp ⊑ Bq iff B˜ p ⊑ B˜ q iff B˜ q (p) = l. Since, by property (ii), Bq (p) ̸ = l implies Bq (p) = 0, this completes the proof. □ We observe that any discriminative and granular polytomous structure satisfying Conditions (i)–(iv) of Theorem 10 is an ordinal polytomous space. It is worth observing that, in the dichotomous case |L| = 2, all the structures on a given set Q of items are level-free and have independent levels. It is only with |L| > 2 that we can separate the class of all the structures satisfying the two properties from that of all the others. All the structures that fall in the former class present a kind of regularity, which is not shared by the structures that are outside the class: (i) The order on the set Q of items is equally represented across all levels, and (ii) within each level l ∈ L+ , atoms B ∈ B(l) have the regular form B(q) > 0 H⇒ B(q) = l.

9

Definition 12. A polytomous structure (Q , L, K) is regular if it is level-free and it has independent levels. Hence, a class hierarchy can be formed: The class of all the structures on the pair (Q , L) includes the class of the ⨆regular d structures on (Q , L), which includes that of the - and -closed regular structures, which is in a one-to-one correspondence with the family of all the quasi-orders on Q . Thus, if the analysis is confined to the regular structures only, then Birkhoff’s Theorem generalizes from the dichotomous to the polytomous case in the obvious way, and the strictly closed structures are just regular structures that happen to be closed under both the infimum and the supremum. 3.5. Beyond quasi-orders: Polytomous surmise systems and spaces Concerning the bijection between surmise relations on a set Q and quasi ordinal polytomous spaces on Q and L, it is worth noticing that so far a ‘‘crisp’’ concept (the dichotomous surmise relation) was mapped to a ‘‘fuzzy’’ one (the quasi ordinal polytomous space) by the bijection. In this section, the situation in which both concepts are ‘‘fuzzy’’ is examined. The notions are further generalized here, by shifting from surmise relations to surmise systems. The theory of knowledge spaces goes beyond the representation of quasi-order relations. Since in the initial development of the theory it was recognized that, while closure under union is a rather reasonable property (two students interacting for a while, will end up to merge their initial knowledge states into a single one which is the union of the two), closure under intersection is a too strict and unnecessary assumption. The structures that are closed under union, but not necessarily under intersection – known as the knowledge spaces – do not represent quasi-orders anymore. Nonetheless, it can be shown that they are in a one-to-one correspondence with another tool for specifying dependencies among items, known as the surmise system (see Definition 1). In the dichotomous theory, surmise systems generalize quasiorder relations in a very precise sense: When the knowledge space is closed under intersection (i.e., it is a quasi-ordinal space), every item has only one clause (i.e., σ (q) is a singleton), and a quasi order ≾ for Q can be derived from σ by setting, for every pair p, q ∈ Q , p ≾ q whenever p is an item in a clause for q. We recall from Definition 3 that in the polytomous framework ⨆ a polytomous space on the pair (Q , L) is a -closed polytomous structure K (notice that we do not require regularity at this stage). The generalization of the surmise system to polytomous items follows. Definition 13. Given a set Q of items and a complete lattice Q (L, ⩽), let µ : Q → 2(L ) be a function mapping items to families of mappings C : Q → L. Then µ is a polytomous attribution on the set Q if µ(q) ̸ = ∅ for all q ∈ Q . Furthermore, if the additional conditions: (1) C ∈ µ(q) implies C (q) ∈ L+ ; (2) C (q′ ) > 0 for some C ∈ µ(q) implies C ′ ⊑ C for some C ′ ∈ µ(q′ ); (3) C , C ′ ∈ µ(q), C (q) = C ′ (q), and C ′ ⊑ C imply C = C ′ hold true for all C , C ′ ∈ LQ and all q, q′ ∈ Q , then µ is named a polytomous surmise function, and the triple (Q , L, µ) a polytomous surmise system. The following theorem is an adaptation to the polytomous framework of Theorem 3.10 in Doignon and Falmagne (1999).

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Theorem 11. There is a one-to-one correspondence between the collection of all the granular polytomous spaces on the pair (Q , L) and the collection of all the polytomous surmise functions on (Q , L). It is defined, for all the granular polytomous spaces K and polytomous surmise functions µ by the formula: C is an atom at q in K ⇐⇒ C is a clause for q in µ, where C : Q → L and q ∈ Q . Proof. Let K⊔ denote the collection of all the granular polytomous spaces on the two sets Q and L, let M denote the collection of all the polytomous attributions for Q and L, and M⊔ ⊆ M be the collection of all the polytomous surmise functions for Q and L. Define the function m : K⊔ → M such that, for K ∈ K and µ ∈ M, m(K) = µ ⇐⇒ ∀q ∈ Q : µ(q) = {C ∈ K : C is an atom at q}. We have to show that m is a bijection between K⊔ and M⊔ . Let K ∈ K⊔ and µ = m(K). By construction, for q ∈ Q , any C ∈ µ(q) is an l-atom at q for some l ∈ L+ . Hence C (q) ∈ L+ and thus µ satisfies Condition (1) of Definition 13. For q, q′ ∈ Q , i ∈ L+ , and C ∈ µ(q), suppose C (q′ ) = i. If C is i-atom at q′ then Condition (2) of Definition 13 trivially holds. Otherwise, if C is not i-atom at q′ , then there is C ′ ∈ K which is i-atom at q′ – thus belonging to µ(q′ ) – and such that C ′ ⊑ C . Hence Condition (2) holds. If C , C ′ ∈ µ(q) and C (q) = C ′ (q) = l ∈ L+ then C and C ′ are both l-atoms at q. Hence they must be incomparable w.r.t. ⊑. Thus Condition (3) of Definition 13 holds. We conclude that m(K) is in M⊔ for all K ∈ K⊔ . Any two spaces K, K′ ∈ K⊔ such that K ̸ = K′ must have distinct collections of atoms B ̸ = B′ . Hence m(K) ̸ = m(K′ ), entailing that m is injective. Let µ ∈ M⊔ and define B=

Fig. 2. Hasse diagram of the polytomous space (Q , L, K8 ), with Q = {a, b, c }, and L = {0, 1, 2}. A state K ∈ K8 such that K (a) = x, K (b) = y, K (c) = z is represented by the ordered triple xyz. This space is not regular.

Proof. Let KR denote the collection of all the regular polytomous spaces on the pair (Q , L), and Σ denote the collection of all the dichotomous surmise systems on Q . Define the function s : KR → Σ by setting, for all K ∈ KR and all σ ∈ Σ , s(K) = σ ⇐⇒ ∀q ∈ Q : σ (q) = {δl (C ) : C is an l-atom

⋃ {µ(q) : q ∈ Q }.

From Condition (3) of Definition 13 it immediately follows that every single element in B is an l-atom at some item q ∈ Q . Hence m is surjective. □ Example 8. For Q = {a, b, c } and L = {0, 1, 2}, the polytomous structure displayed in Fig. 2 is a space (i.e., closed under the supremum). The polytomous surmise function µ8 corresponding to this space is obtained by setting µ8 (q) = {K ∈ K8 :K is an atom at q} for each item q ∈ Q . Hence we have:

µ8 (a) = {100, 200}, µ8 (b) = {010, 020}, µ8 (c) = {101, 011, 222}. It is worth observing that this space has l-bases B8 (1) = {100, 010, 101, 011},

at q for some l ∈ L+ }. Suppose that K, K′ ∈ KR are distinct regular spaces with σ = s(K) and σ ′ = s(K′ ). Then K and K′ must have different l-bases. In particular, since the two spaces are level-free, for any l ∈ L+ there must be q ∈ Q such that the set Aq of all the l-atoms at q in K is different from the set A′q of all the l-atoms at q in K′ . Thus, without loss of generality, there is C ∈ Aq \ A′q . Furthermore, because K and K′ have independent levels, δl (C ) ̸ = δl (C ′ ) for all C ′ ∈ A′q , entailing that σ (q) ̸ = σ ′ (q) and, by this, σ ̸ = σ ′ . Thus s is injective. To show ⋃ that s is also surjective, take +any σ ∈ Σ , define the set B = define the q∈Q σ (q), and for every l ∈ L l-basis B(l) by B(l) = {γl (B) : B ∈ B},

where γl : B → B(l) is such that, given B ∈ B, and F ∈ LQ , F = γl (B) iff

{

l

if q ∈ B

0

if q ∈ Q \ B.

B8 (2) = {200, 020, 222}.

F (q) =

Clearly the two bases are not isomorphic, hence the space is not level-free and thus it is not regular.

Since no element in B can be obtained as the union of other elements in B, there is no element in B(l) that can be obtained as ⋃ the supremum of other elements in B(l). Therefore the set l∈L+ B (l) forms the basis of some regular space K with s(K) = σ. □

The following theorem establishes that only regular spaces have a characterization in terms of dichotomous surmise systems. Thus they correspond to the closed structures of Schrepp (1997). Theorem 12. There is a one-to-one correspondence between the collection of all the granular regular polytomous spaces on the pair (Q , L) and the collection of all the dichotomous surmise functions on Q . It is defined, for all the granular regular polytomous spaces K and dichotomous surmise functions σ as follows. Given l ∈ L+ , C ∈ LQ , q ∈ Q , and a mapping δl (C ) = {q ∈ Q : C (q) = l}, C is an l-atom at q in K ⇐⇒ δl (C ) is a clause for q in σ .

Example 9. For Q = {a, b, c } and L = {0, 1, 2}, the polytomous structure K9 displayed in Fig. 3 is a regular space. The polytomous surmise function corresponding to this space is defined by

µ9 (a) = {100, 200}, µ9 (b) = {010, 020}, µ9 (c) = {101, 011, 202, 022}.

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

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Table 2 Scores of the items of the Rosenberg self-esteem scale (in decreasing order), and atoms of the quasi-ordinal space Kplausible .

Fig. 3. Hasse diagram of the regular polytomous space (Q , L, K9 ), with Q = {a, b, c }, and L = {0, 1, 2}. A state K ∈ K9 such that K (a) = x, K (b) = y, K (c) = z is represented by the ordered triple xyz.

The l-bases of the space are: B9 (1) = {100, 010, 101, 011}, B9 (2) = {200, 020, 202, 022}.

By looking at the two l-bases, it can be easily verified that this space is regular. The corresponding dichotomous surmise system is defined by:

σ9 (a) = {{a}} σ9 (b) = {{b}} σ9 (c) = {{a, c }, {b, c }}. An interesting observation arises at this point. It is well-known that a bijective correspondence exists between the family of all the dichotomous knowledge spaces on a set Q and that of all the dichotomous surmise functions for Q (Doignon & Falmagne, 1999). By Theorem 12 this means that a bijective correspondence also exists between the family of all the dichotomous spaces on Q and that of all the regular polytomous spaces on Q and L. This consideration suggests an alternative characterization of the regular polytomous spaces as those particular polytomous spaces that can be set in a bijective correspondence with the dichotomous spaces. Such a characterization will not be pursued in the present paper.

Item

Score

Rank in the linear order

Levels 1

2

3

i c d j g a f e b h

5934 4883 4875 4698 4687 4608 4427 4048 3709 3635

1 2 3 4 5 6 7 8 9 10

0000000010 0010000010 0011000010 0011000011 0011001011 1011001011 1011011011 1011111011 1111111011 1111111111

0000000020 0020000020 0022000020 0022000022 0022002022 2022002022 2022022022 2022222022 2222222022 2222222222

0000000030 0030000030 0033000030 0033000033 0033003033 3033003033 3033033033 3033333033 3333333033 3333333333

The RSES is often believed to be unidimensional (see, e.g., Fleming & Courtney, 1984; Gray-Little, Williams, & Hancock, 1997; McKay, Boduszek, & Harvey, 2014; Pullmann & Allik, 2000). Accordingly, a Rasch rating scale model (Andrich, 1978) analysis of our data supported the unidimensionality of the scale: The item infit and outfit statistics were lower than 1.5, suggesting that there is no substantial secondary dimension (Smith, 2002). Thus, the relation among the 10 items of the scale can be represented by a linear order. A common way for empirically establishing a linear order on a set of polytomous items q ∈ Q consists of ordering them according to their ‘‘item raw scores’’ Sq , each of which is the sum, across all individuals, of the numerical values assigned to the response categories (de Ayala, 2013). Such a practice can be criticized in that the numbers assigned to the levels in an ordinal scale are arbitrary (provided that the order is respected). In the present application example there was no theoretical knowledge for establishing a polytomous structure on the set of items at issue. Moreover, so far there seem to exist no procedures or methods for building a polytomous structure from empirical data. In this situation, although item raw scores have no valid theoretical foundations, they seem a possible starting point for arriving at some hypothesis about the order on the items. A linear order on the set of ten items of the RSES is obtained as follows. Given any two items q, r ∈ Q with scores Sq and Sr , q ≾ r H⇒ Sq ≥ Sr . Let Q = {a, b, . . . , j} denote the set of the items of the RSES, where a, b and j represent the first, the second and the tenth item of the scale, as they are sorted in the questionnaire. The linear order i≾c≾d≾j≾g ≾a≾f ≾e≾b≾h

4. Empirical example In this section, we apply polytomous KST to a rather wellknown scale in the psychological literature, the Rosenberg SelfEsteem Scale (RSES; Rosenberg, 1965), for which a linear order on the items is usually assumed. The aim of the application is to show and underline the differences that arise when comparing the polytomous KST approach to ‘‘standard’’ ones. The data considered below consists of the responses that 2251 individuals (mean age = 17.34 ± 7.56 years; 60.1% females) gave to the RSES. The scale consists of 10 items rated on a 4point agreement scale from 0 (Strongly Disagree) to 3 (Strongly Agree). Five items are positively worded (i.e., the stronger the agreement, the higher the self-esteem), and five are negatively worded (i.e., the stronger the agreement, the lower the selfesteem). The responses to the latter items were rescored prior to analyses.

(3)

was obtained on the basis of the item raw scores (Table 2). Let L = {0, 1, 2, 3} be the set of the four response levels of the RSES. There are 286 belief states that are compatible with the linear order in (3). The collection of these states is the quasi-ordinal space (Q , L, Kplausible ) displayed in Fig. 4. For illustrative purposes, Fig. 5 shows the 20 belief states that are obtained on the domain restriction Q ′ = {a, b, c }, with c ≾ a ≾ b. It is worth noting that the space in Fig. 5 is isomorphic to the space obtainable by any choice of three items x, y, z ∈ Q such that x ≾ y ≾ z. As it can be observed from the two figures, although both the items in Q and Q ′ , and the levels in L are linearly ordered, the states are only partially ordered by ⊑. The space Kplausible has 30 different atoms, which are listed in Table 2. This space has three l-bases, one for each of the levels in {1, 2, 3}. Each l-basis is the collection of all the atoms that are listed in the column l of the

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Fig. 5. Hasse diagram of the quasi-ordinal space (Q ′ , L, K′plausible ), with Q ′ = {a, b, c }, and L = {0, 1, 2, 3}. K′plausible is the collection of all the states that are compatible with the linear order c ≾ a ≾ b.

Q Define ∑the ‘‘individual raw score’’ of an element K ∈ L by K+ = K (q). In a standard psychometric approach, two q∈Q different belief states K1 and K2 having the same individual score K1+ = K2+ would endorse the same ‘‘level of self-esteem’’ for the individuals in those states. However, there are differences among the states having the same individual score that could be informative from a psychological perspective. For instance, the two belief states

K1 = 1122112122, and K2 = 0033003033

Fig. 4. Hasse diagram of the quasi-ordinal space (Q , L, Kplausible ), with Q = {a, b, . . . , j}, and L = {0, 1, 2, 3}. Kplausible is the collection of all the states that are compatible with the linear order in (3). The Hasse diagram reads from top (infimum state) to bottom (supremum state).

table. It can be easily checked that the three l-bases are pairwise isomorphic and hence the space is level-free. By direct inspection it can also be checked that the space has independent levels.

in Kplausible have an individual score of 15. However, the former could be the state of an individual who chooses the moderate response categories (1 = Disagree, 2 = Agree), whereas the latter could be the state of an individual who chooses the extreme response categories (0 = Strongly Disagree, 3 = Strongly Agree). These differences would be hidden if only the individual score is considered. As pointed out (for the dichotomous items case) by Bottesi et al. (2015), Serra, Spoto, Ghisi, and Vidotto (2017) and Spoto, Stefanutti, and Vidotto (2010), this issue can be very critical when questionnaires are used, for instance, in psychological assessment to define a diagnosis and, therefore, to implement the treatment. It was interesting to empirically validate the polytomous structure Kplausible by using the data at hand. Many probabilistic models are available for the empirical validation of dichotomous

L. Stefanutti, P. Anselmi, D. de Chiusole et al. / Journal of Mathematical Psychology 94 (2020) 102306

13

obtained in the following way: R ∆′ K =

∑

|R(q) − K (q)|.

q∈Q

This distance is equivalent to a Manhattan distance. In general, the distance R∆′ K goes from 0 to the largest integer smaller or (|L|−1)×|Q | . equal to 2 The distance of a response pattern R ∈ R from a polytomous structure K is then computed as d(R, K) = min{R∆′ K : K ∈ K}.

Fig. 6. Discrepancy distribution from the data to the plausible space (gray bars) and average of the 100,000 discrepancy distributions from the data to the alternative spaces (black bars).

knowledge structures (de Chiusole & Stefanutti, 2013; de Chiusole, Stefanutti, Anselmi, & Robusto, 2015a; Falmagne & Doignon, 1988a, 1988b; Robusto, Stefanutti, & Anselmi, 2010; Stefanutti, Anselmi, & Robusto, 2011). Unfortunately, such models are not (directly) applicable in the polytomous case. Developing or extending models of such kind is beyond the scope of the present article. Nonetheless, an index of the distance between a structure and the data was obtained as described below. Although such an index is far from being a fit statistic, it provides a simple way of comparing a structure with observed data. Of all the linear orders on the set Q , one could look for the one that best fits the data. A brute force method would consist of evaluating each of the 10! possible linear orders on a set of 10 items. In this example, a total of 100,000 alternative linear orders were randomly sampled without replacement from the 10! − 1 permutations of the 10 items that were different from the plausible linear order in (3). A quasi-ordinal space (Q , L, Kx ), with L = {0, 1, 2, 3} and x = {1, 2, . . . , 100, 000}, was obtained for each alternative linear order. All the alternative spaces Kx had the same cardinality of Kplausible (i.e., 286 belief states). For denoting the data set, let F : LQ → R+ be the function assigning observed frequencies to elements in LQ . Moreover let R = {R ∈ LQ : F (R) > 0} be the collection of the observed elements in LQ , that in the sequel are named response patterns. A measure of the distance between a polytomous structure K and a data set R can be obtained in, at least, two different ways. Given a response pattern R ∈ R and a state K ∈ K, a direct generalization of the symmetric difference distance, as computed in the dichotomous case, is obtained by defining, for all q ∈ Q

δq (R, K ) =

{

0

if R(q) = K (q),

1

if R(q) ̸ = K (q),

where R(q) (respectively, K (q)) is the level l ∈ L of item q ∈ Q in response pattern R (respectively, state K ). Then, the symmetric difference distance is equivalent to a Hamming distance: R∆K =

∑

δq (R, K ).

q∈Q

This distance does not take into account the full information about the ordinal nature of the responses but treats responses as if they were measured at a nominal level. An alternative metric, which is appropriate for the ordinal nature of the responses, is

Performing this computation for all response patterns of R, and summing the frequencies F (R) of all patterns having the same minimum distance d(R, K), a frequency distribution of the minimum distances is obtained. This distribution is referred to as the discrepancy distribution from the data set R to the polytomous structure K. A discrepancy index from R to K is obtained by computing the mean of the discrepancy distribution from R to K: D(R, K) =

∑

R∈ R

∑

d(R, K)F (R)

R∈R

F (R)

.

D(R, K) can be regarded as a measure of how well a polytomous structure K fits a data set R. The lower the value of D(R, K), the better the fit of K to R. It is worth noting that D(R, K) = 0 whenever R ⊆ K. Fig. 6 shows the discrepancy distribution from the data to the plausible space (gray bars), and the average of the 100,000 discrepancy distributions from the data to the alternative spaces (black bars). In the figure, the frequencies of the minimum distances have been converted into relative frequencies by dividing them by the number of individuals. The observed minimum distances ranged from 0 to 15, which is the maximum possible distance in the case of 10 items with 4 response levels. Minimum distances less than or equal to 3 were more frequent in the plausible space than in the alternative spaces, whereas minimum distances larger than 3 were more frequent in the alternative spaces. The discrepancy index from the data to the plausible space was 2.86. The discrepancy indexes from the data to the alternative spaces ranged from 2.81 to 4.76, with an average value of 4.10. The discrepancy index of the plausible space was almost 4 standard deviations lower than the average discrepancy index of the alternative spaces (Cohen’s d = −3.78). Only five alternative spaces (out of 100,000) fit the data better than the plausible space (i.e., their discrepancy index was less than 2.86). In Fig. 7, the Spearman’s correlation between the item rankings of the plausible and alternative linear orders (x-axis) is plotted against the difference between the discrepancy indexes of the alternative and plausible spaces (y-axis). The more the alternative linear orders were dissimilar from the plausible linear order (ρ close to −1), the worse the alternative spaces fit the data. The more the alternative linear orders were similar to the plausible linear order (ρ close to 1), the more the fit of the alternative spaces to the data approached that of the plausible space. The five alternative spaces that fit the data better than the plausible space (the five dots below the dashed line) were based on linear orders which largely resembled the plausible linear order (ρ from .92 to .99). 5. Discussion The aim of the present article was to provide the mathematical foundation for the generalization of KST to the case of more than two ordered response categories. The main idea was addressed by Schrepp (1997) twenty years ago. In a pioneering article, he found the condition under which a correspondence between

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Fig. 7. Correlation between the item rankings of the plausible and alternative linear orders (x-axis) against the difference between the discrepancy indexes of the alternative and plausible spaces (y-axis).

surmise functions and knowledge spaces exists when multiple, rather than dichotomous, answer alternatives are admitted. In the present article, the question of whether strict closure can be decomposed into a necessary and sufficient set of independent properties was addressed. In particular, it is proved that a structure is strictly closed if the following four different conditions hold: It is level-free, it has independent levels, and it is ⊔- and ⊓-closed (see Theorem 10). The level-free condition establishes that a certain order on the set of items is preserved across all the levels of the rating scale. Level independence requires that the l-atoms assign to each item either l or 0. A structure satisfying these two conditions is named regular. Concerning the more general collection of all the (regular and non-regular) structures, the existence of a bijective correspondence between the family of all the polytomous spaces and the polytomous surmise functions was provided. This shows that, through the decomposition of the closure condition into four independent conditions, polytomous KST can be approached from a more general perspective. The entire article deals with the deterministic mathematical foundation of the polytomous KST, but to be able to apply it in practice, as recently discussed also by Ünlü et al. (2013), some further questions have to be solved. First, the probabilistic counterpart allowing the empirical validation and application of the structures has to be developed. One possible direction can be that of extending the basic local independence model (BLIM; Falmagne & Doignon, 1988a, 1988b) to the polytomous case, by assuming a careless error and a lucky guess parameter for each level of each item of the domain. Once it is done, it will be possible to generalize all the extensions of the BLIM for dichotomous data (Anselmi, Robusto, & Stefanutti, 2012; Anselmi, Stefanutti, de Chiusole, & Robusto, 2017; de Chiusole, Anselmi, Stefanutti & Robusto, 2013; de Chiusole, Stefanutti, Anselmi & Robusto, 2013; de Chiusole et al., 2015a; de Chiusole, Stefanutti, Anselmi, & Robusto, 2015b; Heller & Wickelmaier, 2013; Robusto et al., 2010; Stefanutti et al., 2011; Stefanutti & de Chiusole, 2017). Second, some methods for building polytomous structures have also to be developed. As far as expert query is concerned, a theorem by Koppen and Doignon (1990) establishes a bijective correspondence between knowledge spaces and a special type of binary relations known as entail relations. What the expert builds

in practice is the entail relation. The space is indirectly obtained from this last. At the moment, this type of correspondence is still missing in the polytomous case. Future work could be devoted to this purpose. In another direction, it should not be so difficult to extend to the polytomous case many data-driven construction procedures such as the inductive item tree analysis (IITA; Sargin & Ünlü, 2009; Ünlü & Sargin, 2010), the data-driven skill map extraction procedure (D-SMEP; Spoto, Stefanutti, & Vidotto, 2016), the k-states procedure (de Chiusole, Stefanutti, & Spoto, 2017), and still other methods (Robusto & Stefanutti, 2014; Schrepp, 1999). Third, some adaptive assessment procedures, both at the deterministic and the probabilistic levels, can be developed, along the lines of Degreef, Doignon, Ducamp, and Falmagne (1986) and Falmagne and Doignon (1988a, 1988b); see also Anselmi, Robusto, Stefanutti, and de Chiusole (2016) and Hockemeyer (2002). In particular, the probabilistic procedures will rely on the probabilistic developments of the polytomous theory that have been mentioned above. There are connections between the topic of the present article and those in other research areas. One of them is a well studied object in combinatorial convexity. In case the order (L, ⩽) is the real unit interval [0, 1] with its usual ordering, the collection of states in an ordinal polytomous structure is nothing else than the order polytope, which is investigated, for instance, in Stanley (1986). These connections will be explored in future studies. Acknowledgments We are very grateful to Jean-Paul Doignon for his constructive and helpful suggestions and comments to a previous version of the manuscript. We also thank Carlo Chiorri for having made available to us the data set used in the empirical application of this article. The research developed in this article was carried out under the research project CPDR152105 ‘‘Learning how students learn. Mathematical modeling of learning processes in intelligent tutoring system navigation’’, funded by the FISPPA Department, University of Padua, Italy. References Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573. Anselmi, P., Robusto, E., & Stefanutti, L. (2012). Uncovering the best skill multimap by constraining the error probabilities of the gain-loss model. Psychometrika, 77(4), 763–781. Anselmi, P., Robusto, E., Stefanutti, L., & de Chiusole, D. (2016). An upgrading procedure for adaptive assessment of knowledge. Psychometrika, 81(2), 461–482. Anselmi, P., Stefanutti, L., de Chiusole, D., & Robusto, E. (2017). The assessment of knowledge and learning in competence spaces: The gain–loss model for dependent skills. British Journal of Mathematical and Statistical Psychology, 70(3), 457–479. Bartl, E., & Belohlavek, R. (2011). Knowledge spaces with graded knowledge states. Information Sciences, 181(8), 1426–1439. Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal, 3, 443–454. Bottesi, G., Spoto, A., Freeston, M. H., Sanavio, E., & Vidotto, G. (2015). Beyond the score: Clinical evaluation through formal psychological assessment. Journal of Personality Assessment, 97(3), 252–260. Davey, B. A., & Priestley, H. A. (2002). Introduction to lattices and order (2nd ed.). Cambridge, UK: Cambridge University Press. de Ayala, R. J. (2013). The theory and practice of item response theory. New York: The Guilford Press. de Chiusole, D., Anselmi, P., Stefanutti, L., & Robusto, E. (2013). The Gain–Loss Model: Bias and variance of the parameter estimates. Electronic Notes in Discrete Mathematics, 42, 33–40. de Chiusole, D., & Stefanutti, L. (2013). Modeling skill dependence in probabilistic competence structures. Electronic Notes in Discrete Mathematics, 42, 41–48. de Chiusole, D., Stefanutti, L., Anselmi, P., & Robusto, E. (2013). Assessing parameter invariance in the BLIM: Bipartition models. Psychometrika, 78, 710–724. http://dx.doi.org/10.1007/s11336-013-9325-5.

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