This paper combines two aspects in the construction of knowledge spaces. The first is to utilize a partial order of skills which belong to a given domain of knowledge to find items or problems which finally constitute a knowledge space or a learning space. Secondly the orders of skills of different (related) areas are coalesced by well known order operations to construct larger spaces. In the first part a so called basic set representation of an order is utilized to considerably simplify the construction of knowledge spaces. In the second part the techniques to "add'' and "multiply'' orders are applied to given orders resulting in new orders to which the set representation theory of the first part is applied. Properties of the building blocks are identified which can be transferred to the combination and yield properties of the final knowledge space which make their handling a bit more convenient. In particular, places which proved useful in the construction of the first part are identified when these order operations are utilized. (C) 2021 Elsevier Inc. All rights reserved.

The structure of the set of all possible rating scales is investigated. It is shown that by a natural addition of rating scales the set is a commutative semigroup with neutral element. From this operation a partial order can be defined which turns out to a lattice order. This lattice is shown to be distributive. In the next step two possibilities –closely related to the preceding development –are analyzed to endow this structure with a metric. The semigroup operation is shown to be continuous in the respective topologies. With the help of one of these metrics the question of the scale type of rating scales is discussed by giving the concept of admissible transformations an extended meaning.

It has been mentioned on many occasions that Formal Concept Analysis and KST, the theory of Knowledge Spaces, introduced by J.-P. Doignon and J.-C. Falmagne, are closely related in theory, but rather different in practice. It was suggested that the FCA community should learn from and contribute to KST. In a recent workshop held at Graz University of Technology, researchers from both areas started to combine their views and tried to find a common language. This article is a partial result of their effort. It invites FCA researchers to understand some ideas of KST by presenting them in the language of formal contexts and formal concepts.

Regular choice systems and their random utility representations are investigated. A generalization of the derivation of the Block–Marschak conditions, based on the Möbius function of a partial order is presented. The technique is demonstrated in connection with two examples. The first is similar to complete choice data. In the second example a complete characterization of the ensuing polytope is obtained including a procedure to explicitly derive a convex representation of a data matrix if it is in the polytope.

In computerized assessment of knowledge it is important to quickly estimate the competence state of a testee. This is particularly true for digital educational games where this kind of assessment has to be done in a non-invasive way, i.e., by avoiding any queries or interruptions. This paper presents the mathematical foundation of a model by which a large set of competence states is partitioned in disjoint subsets and the probability of a subject being in a particular state is calculated for the subsets of states and updated according to the monitored performance. Based on this calculation adaptive interventions can be automatically chosen by tailoring the upcoming problem to the competence state of the user.

The concepts of a relation, a mapping, and an operation are introduced. An ordered relational structure consists of a set and a few relations defined on it. The formal definition is presented and illustrated by numerous examples from social sciences and from mathematics. Homomorphisms and isomorphisms between structures are defined. These are mappings of one structure into another which, in a way, keep its structural properties, while possibly changing the meaning of its primitives. Automorphisms are mappings of the structure onto itself. They reveal most of its important properties. The group of automorphisms of a structure is described leading to the concept of homogeneity and uniqueness of a structure. The Alper/Narens theorem characterizes structures on the reals with specified homogeneity and uniqueness properties of their automorphism group. This famous theorem is discussed along with a recent generalization. The order topology and some results in connection with it are presented. Finally, the problems which arise when a structure is embedded into a Dedekind complete ordered relational structure, are briefly mentioned. This article is closely related to the article on representational measurement theory. Furthermore, the section on the completion of structures draws on Section Order Topology of the article Partial Orders.

We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

Competence-based Knowledge Space Theory (CbKST) has been proven to be a very well-fitting basis for realizing personalization in technology-enhanced learning. Especially in the area of game-based learning, however, some extensions and improvements are needed.Personalization in a serious game cannot be regarded simply as the selection of game assets according to the individual learner's current competences but it must also pay heed to the up-keeping of a storyline, it must be ensured that no part of the story is omitted that may be necessary to understand a later part. Therefore, a CbKST-compatible Markovian model for storytelling is proposed.A second issue is the ongoing, non-invasive assessment of the learner's current competences during the game. Every action of the learner within the game should be taken into account for the competence assessment, and the assessment must be done in real-time, i.e. there must not be any delay caused by the assessment which would interrupt the flow of the game. A simplified update procedure for competence assessment within CbKST is suggested which can solve this issue, and simulation results are presented comparing the new procedure with the classical one.

Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders); their characteristic vectors are regarded as points in a Euclidean space; their convex hull forms a polytope whose facets are fully characterized. This polytope is shown to correspond to an order polytope. Furthermore, its relation to the biorder polytope is pointed out. The convex representations of a given point of the polytope are discussed. The impact of these results on the methods of analyzing data arising from a categorical judgement procedure is outlined. In particular, some consequences are drawn with respect to the usual evaluation of correlations of such data.

Set representations are useful in the theory of knowledge spaces. A set representation of an order is an isomorphic mapping of its base set into the power set of some set ordered by set inclusion. Such a representation is basic if the union of the representing sets of the predecessors of an element contains strictly less elements than the representing set of this element, and it is parsimonious if the difference is exactly one element. This paper investigates the properties of the minimal number of elements which must be used in a parsimonious representation. This value is studied for several order operations. Moreover, orders which allow essentially only one parsimonious set representation are structurally characterized. These orders are called saturated. Finally, the way to apply these results to knowledge spaces are outlined.

Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds of such set representations are investigated because they facilitate the exploration of properties of knowledge spaces. A new type of order relation is defined which is closely related to interval orders. These orders and the interval orders are investigated with respect to their parsimonious set representations. The theorems are applied to knowledge space theory.

The question of the existence of independent random variables which represent a given set of binary choice data is investigated. It is related to the linear ordering polytope. The subset of it which is independently representable, denoted by ILOn, is characterized for three elements. For the general case, necessary conditions are given and their geometric meaning is discussed. A procedure, called mixture technique, is developed which allows one to construct a new point in ILOn and its independent representation from known points in ILOn. Finally, a few results on parametric representations are derived.

The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed. Copyright 2001 Academic Press.