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May 2011
Hermann Rodenhausen
Knowledge Description and Galois Correspondence
Practical Impact of a Structural Idea
The concept of a Galois connection has shown to be an important tool in the analysis of relations between various kinds of order structures. Implicitly, it was used by Galois in his theory of algebraic equations: given fields K and F, F extending K, the relation between the subgroups of the Galois group G(F, K) and the fields intermediate between K and F can be adequately described by a Galois connection. Another example of a Galois connection arises in logic by relating structures and sentences via the validity relation. Within computer science and formal linguistics, a well-known example of a Galois connection is given by the notion of a "context" in formal concept analysis. This model results in the general construction of Galois connections out of binary relations - interpreted as "incidence" between objects and attributes. The paradigmatical character and generality of this model give rise to the question how much the concept of a Galois connection actually differs from the underlying concept of a binary relation; in this book, this question is addressed by proving a representation theorem being more general than some of the representation results known from the literature.
Any Galois connection comprises natural closure operators on its components; in many classical examples, these closure operators have appealing algebraic descriptions. In this book the principles determining the outcome of the closure are analyzed by using methods of logic; in this way, a relation between structural operations (closure concepts) from different algebraic categories can be established. Actually, it is shown that there is an intrinsic relation between forms of algebraic closure and the concept of logical entailment known from propositional logic.
Further results concern the concept of a "basis" in the context of partial orders combining both, aspects of generative power and minimality. It is shown that bases generally exist and are uniquely determined. Moreover, structural aspects characterizing them can in convenient ways be described by a Galois connection.
It is well-known that Galois connections are strongly related to the psychological theory of "knowledge spaces", a conceptual framework that has recently been applied to the development of intelligent tutorial systems. We extend the focus of application by showing that knowledge space theory and tools used for its mathematical description can particularly be adapted to the study of mathematical misconceptions. We show that - in the domain considered - forms of conceptual failure display a high measure of regularity - a fact that parallels observations known from elementary algebra.
The research presented here shows that analyzing empirical data and solving conceptual problems associated to the type of application may relate in significant ways to mathematical questions of non-trivial character.
Any Galois connection comprises natural closure operators on its components; in many classical examples, these closure operators have appealing algebraic descriptions. In this book the principles determining the outcome of the closure are analyzed by using methods of logic; in this way, a relation between structural operations (closure concepts) from different algebraic categories can be established. Actually, it is shown that there is an intrinsic relation between forms of algebraic closure and the concept of logical entailment known from propositional logic.
Further results concern the concept of a "basis" in the context of partial orders combining both, aspects of generative power and minimality. It is shown that bases generally exist and are uniquely determined. Moreover, structural aspects characterizing them can in convenient ways be described by a Galois connection.
It is well-known that Galois connections are strongly related to the psychological theory of "knowledge spaces", a conceptual framework that has recently been applied to the development of intelligent tutorial systems. We extend the focus of application by showing that knowledge space theory and tools used for its mathematical description can particularly be adapted to the study of mathematical misconceptions. We show that - in the domain considered - forms of conceptual failure display a high measure of regularity - a fact that parallels observations known from elementary algebra.
The research presented here shows that analyzing empirical data and solving conceptual problems associated to the type of application may relate in significant ways to mathematical questions of non-trivial character.
Keywords: quasi-orders; Galois connections; closure conditions; knowledge spaces; model theory
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