Learning Spaces – Jean – Claude Falmagne (1)

(10.13) 10.4.11 Theorem. For any discriminative, well-graded family K of sets, de- note by s(K) the oriented medium constructed in Theorems 10.4.1 and 10.4.3. For any oriented medium (S, T), denote by f(S, T) the discriminative, well- graded family formed by all its positive contents (cf. Theorem 10.4.9). We then have K ∼(f ◦s)(K) and S ∼(s ◦f)(S). (10.14) Proof. Given K, select some q ∈∪K \ ∩K and some K ∈K. Note that K stands for both an element of K and an element of s(K), since these two sets are equal.

We have q ∈K if and only if τq ∈bK+. With (S, T) = s(K), the positive tokens of the oriented medium (S, T) are of the form τq, for q in ∪K \ ∩K. Also, f(S, T) is the family bS+ of all positive contents of (S, T), and any positive token τ is an element of ∪bS+; moreover, we have ∩bS+ = ∅by Theorem 10.4.9. The first isomorphism formula in (10.14) derives from the mapping (∪K) \ (∩K) →bS+ : q 7→τq.

Let (S, T) be an oriented medium, and let (S′, T′) denote (s ◦f)(S, T). An element K of the set family K = f(S, T) is exactly the positive content of some state of the medium (S, T). Each state in the medium s(K) = (S′, T′) is an element of K, so is a positive content bS+ of some state S in the given medium (S, T).

By Theorem 10.4.9, S is fully determined by bS+. We may thus set b(S) = K, which gives a bijective mapping b : S →S′. Now, what about the tokens of (S′, T′)? According to Equations (10.7) and (10.8), a token τ of s(K) = (S′, T′) is either of the form τq or τ¯q = eτq, where q ∈(∪K) \ (∩K).

Theorem 10.4.9 gives us here ∪K = ∪bS+ = T+ and ∩K = ∩bS+ = ∅. Then, any q ∈∪K = T+ equals some positive token τ(q) of the given medium (S, T), and conversely. This yields the bijective mapping c : T →T′ sending τ(q) to τq, and g τ(q) to τ¯q.

Springer Heidelberg Dordrecht London New York c⃝Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer.

Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: KuenkelLopka GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Library of Congress Control Number: 2010933254 Prof. Dr. Jean-Paul Doignon Université Libre de Bruxelles Département de Mathématique, c.p.

216 Bd du Triomphe 1050 Bruxelles Belgium [email protected] Prof. Dr. Jean-Claude Falmagne 92697-5100 Irvine California USA [email protected] Department of Cognitive Sciences Institute of Mathematical Behavioral Sciences University of California, Irvine Preface This book is a much enlarged second edition of “Knowledge Spaces”, by Jean- Paul Doignon and Jean-Claude Falmagne, which appeared in 1999. Chapters content are explained below. As much of our earlier preface remains valid, we reproduce the useful parts here. The work reported in these pages began during the academic year 1982– 83. One of us (JCF) was on sabbatical leave at the University of Regensburg.

For various reasons, the time was ripe for a new joint research subject. Our long term collaboration was thriving, and we could envisage an ambitious commitment. We decided to build an efficient machine for the assessment of knowledge—for example, that of students learning a scholarly subject. We be- gan at once to work out the theoretical components of such a machine.

Until then, we had been engaged in topics dealing mostly with geometry, combi- natorics, psychophysics, and especially measurement theory.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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