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Über dieses Buch

D. Hilbert, in his famous program, formulated many open mathematical problems which were stimulating for the development of mathematics and a fruitful source of very deep and fundamental ideas. During the whole 20th century, mathematicians and specialists in other fields have been solving problems which can be traced back to Hilbert's program, and today there are many basic results stimulated by this program. It is sure that even at the beginning of the third millennium, mathematicians will still have much to do. One of his most interesting ideas, lying between mathematics and physics, is his sixth problem: To find a few physical axioms which, similar to the axioms of geometry, can describe a theory for a class of physical events that is as large as possible. We try to present some ideas inspired by Hilbert's sixth problem and give some partial results which may contribute to its solution. In the Thirties the situation in both physics and mathematics was very interesting. A.N. Kolmogorov published his fundamental work Grundbegriffe der Wahrschein­ lichkeitsrechnung in which he, for the first time, axiomatized modern probability theory. From the mathematical point of view, in Kolmogorov's model, the set L of ex­ perimentally verifiable events forms a Boolean a-algebra and, by the Loomis-Sikorski theorem, roughly speaking can be represented by a a-algebra S of subsets of some non-void set n.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
One of the most important problems initiated by quantum theory is a mathematical description of the structure of random events of a quantum mechanical system. This problem was originally formulated in the famous paper by G. Birkhoff and J. von Neumann “The Logic of Quantum Mechanics” in 1936. The fundamental difficulty is the existence of quantum phenomena that cannot be described in terms of the event structure of the classical Kolmogorovian probability theory. It is noteworthy that similar situations seem to arise in other areas of sciences; e.g., psychology, design of computers, neural networks, and the biology of the human brain.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 1. D-posets and Effect Algebras

Abstract
In this Chapter, we first introduce and investigate basic properties of D-posets and effect algebras as partial algebraic structures. We introduce the notions of their morphisms and prove a categorical equivalence of D-posets and effect algebras. We show relations of these structures with partially ordered Abelian groups and prove the existence of a universal group for effect algebras.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 2. MV-algebras and QMV-algebras

Abstract
In this chapter, we study supplement algebras, MV-algebras and QMV-algebras. MV-algebras were introduced by C. Chang [Cha] as algebraic models of many-valued logics. The prototypical model of MV-algebras is based on the real interval [0, 1]. Nowadays, there is a huge literature devoted to MV-algebras. In this chapter, we introduce only their basic properties. For more details see, e.g., the recent monograph [CDM].
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 3. Quotients of Partial Abelian Monoids

Abstract
The basic algebraic structure that is studied in this chapter is a partial Abelian monoid (PAM in short) (cf. [Wil 1], [Wil 2], [Pul 4], [GuPu]). A PAM is a structure (P; 0, ⊕), where e is a commutative, associative partial binary operation on P and 0 is a neutral element. Beginning with a PAM at the lowest level, we shall consider a hierarchy of partial algebraic structures. The second level is a cancellative PAM (CPAM), the third level is a generalized effect algebra/generalized difference poset, which coincide with a cancellative, positive PAM. Commutative positive minimal clans and BCK-algebras are also included. An effect algebra is a unital generalized effect algebra/D-poset. On higher levels in the hierarchy we find orthoalgebras, orthomodular posets and lattices, MV-algebras, Boolean algebras.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 4. Tensor Product of D-Posets and Effect Algebras

Abstract
The event structure of a quantum physical system is identified with a quantum logic [BLM] or an orthoalgebra [FoRa 1], [FoPt] in contrast to classical mechanics when it is assumed to be a Boolean algebra. One of important problems is a coupled system of two independent physical systems P and Q. The event structure L of this coupled system L, if it exists, is usually called a tensor product, and we write L = P ⊗ Q.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 5. BCK-algebras

Abstract
In 1966, Imai and Iséki [ImIs], [Ise] introduced the notion of a BCK-algebra. This notion originated from two different ways: (1) set theory, and (2) classical and non-classical propositional calculi. The BCK-operation * is an analogue of the settheoretical difference. Today BCK-algebras have been studied by many authors and they have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory, topology, fuzzy set theory, and so on.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 6. BCK-algebras in Applications

Abstract
We apply general methods of BCK-algebras to obtain a finer analysis of BCK-algebras to describe semisimple BCK-algebras, simple BCK-algebras, bounded commutative BCK-algebras, to make a comparison with difference posets and to describe pseudo MV-algebras.
Anatolij Dvurečenskij, Sylvia Pulmannová

Chapter 7. Loomis-Sikorski Theorems for MV-algebras and BCK-algebras

Abstract
We recall that σ-complete MV-algebras are MV-algebras which are σ-complete lattices. Such MV-algebras are always semisimple algebras, and they are exactly those for which there exists an MV-isomorphism with a Bold algebra, i.e., with an algebra of fuzzy sets on a crisp set Ω which contains lΩ, and which is closed under the fuzzy complementation and formation of min{f + g, l}. Belluce [Bel] showed that every semisimple MV-algebra M can be always represented as a Bold algebra of continuous fuzzy sets on the compact Hausdorff space of all maximal ideals of M. And this is an analogue of Stone’s representation theorem for Boolean algebras. Situation with σ-complete MV-algebras is more complicated as we will see below.
Anatolij Dvurečenskij, Sylvia Pulmannová

Backmatter

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