Ebook: New Trends in Quantum Structures

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.

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This monograph deals with the latest results concerning different types of quantum structures. This is an interdisciplinary realm joining mathematics, logic and fuzzy reasoning with mathematical foundations of quantum mechanics, and the book covers many applications.
The book consists of seven chapters. The first four chapters are devoted to difference posets and effect algebras; MV-algebras and quantum MV-algebras, and their quotients; and to tensor product of difference posets. Chapters 5 and 6 discuss BCK-algebras with their applications. Chapter 7 addresses Loomis-Sikorski-type theorems for MV-algebras and BCK-algebras. Throughout the book, important facts and concepts are illustrated by exercises.
Audience: This book will be of interest to mathematicians, physicists, logicians, philosophers, quantum computer experts, and students interested in mathematical foundations of quantum mechanics as well as in non-commutative measure theory, orthomodular lattices, MV-algebras, effect algebras, Hilbert space quantum mechanics, and fuzzy set theory.

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This monograph deals with the latest results concerning different types of quantum structures. This is an interdisciplinary realm joining mathematics, logic and fuzzy reasoning with mathematical foundations of quantum mechanics, and the book covers many applications.
The book consists of seven chapters. The first four chapters are devoted to difference posets and effect algebras; MV-algebras and quantum MV-algebras, and their quotients; and to tensor product of difference posets. Chapters 5 and 6 discuss BCK-algebras with their applications. Chapter 7 addresses Loomis-Sikorski-type theorems for MV-algebras and BCK-algebras. Throughout the book, important facts and concepts are illustrated by exercises.
Audience: This book will be of interest to mathematicians, physicists, logicians, philosophers, quantum computer experts, and students interested in mathematical foundations of quantum mechanics as well as in non-commutative measure theory, orthomodular lattices, MV-algebras, effect algebras, Hilbert space quantum mechanics, and fuzzy set theory.
Content:
Front Matter....Pages i-xvi
Introduction....Pages 1-8
D-posets and Effect Algebras....Pages 9-127
MV-algebras and QMV-algebras....Pages 129-189
Quotients of Partial Abelian Monoids....Pages 191-229
Tensor Product of D-Posets and Effect Algebras....Pages 231-291
BCK-algebras....Pages 293-377
BCK-algebras in Applications....Pages 379-446
Loomis-Sikorski Theorems for MV-algebras and BCK-algebras....Pages 447-490
Back Matter....Pages 491-541

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This monograph deals with the latest results concerning different types of quantum structures. This is an interdisciplinary realm joining mathematics, logic and fuzzy reasoning with mathematical foundations of quantum mechanics, and the book covers many applications.
The book consists of seven chapters. The first four chapters are devoted to difference posets and effect algebras; MV-algebras and quantum MV-algebras, and their quotients; and to tensor product of difference posets. Chapters 5 and 6 discuss BCK-algebras with their applications. Chapter 7 addresses Loomis-Sikorski-type theorems for MV-algebras and BCK-algebras. Throughout the book, important facts and concepts are illustrated by exercises.
Audience: This book will be of interest to mathematicians, physicists, logicians, philosophers, quantum computer experts, and students interested in mathematical foundations of quantum mechanics as well as in non-commutative measure theory, orthomodular lattices, MV-algebras, effect algebras, Hilbert space quantum mechanics, and fuzzy set theory.
Content:
Front Matter....Pages i-xvi
Introduction....Pages 1-8
D-posets and Effect Algebras....Pages 9-127
MV-algebras and QMV-algebras....Pages 129-189
Quotients of Partial Abelian Monoids....Pages 191-229
Tensor Product of D-Posets and Effect Algebras....Pages 231-291
BCK-algebras....Pages 293-377
BCK-algebras in Applications....Pages 379-446
Loomis-Sikorski Theorems for MV-algebras and BCK-algebras....Pages 447-490
Back Matter....Pages 491-541
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