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New trends in quantum structures. (English) Zbl 0987.81005
Mathematics and its Applications (Dordrecht). 516. Dordrecht: Kluwer Academic Publishers. Bratislava: Ister Science. xvi, 541 p. (2000).
Study of algebraic quantum structures was initiated in 1936 by G. Birkhoff and J. von Neumann who showed that the structure of the set of assertions on a quantum system is weaker than that of Boolean algebra. This algebraic approach to the logic of quantum systems develops vigorously from the sixties, and results of the development have been reflected in numerous monographs. The authors of the book under review are widely known representatives of the Slovak school and have contributed to the subject enormously.
The book aims at introducing the reader into the realm of the latest results and methods clustering around D-posets and effect algebras, MV-algebras, and BCK-algebras, all combined with ideas of fuzzy set theory. It may be of interest both for physicists and mathematicians, and even for computer scientists working in automata theory, quantum computing or artificial intelligence.
The first of the seven chapters introduces the basic definitions and properties of D-posets and effect algebras, including equivalence of the two concepts and their relation to ordered Abelian groups. Special attention is paid to interval effect algebras and effect algebras with Riesz decomposition property. The second chapter is devoted to MV-algebras and quantum MV-algebras. The latter provide a common abstraction of MV-algebras and orthomodular lattices. Ideals and congruences of structures of both kind are also studied here. Ideals, congruences and dimension theory in weaker structures, partial Abelian monoids, are discussed in Chapter 3. The subject of the next chapter is tensor products of D-posets (and effect algebras): they represent event structures of pairs of independent physical systems. Also logics of test spaces and of partitions, as well as connections between partition logics and automata are described here.
The fifth chapter deals with BCK-algebras, structures which only recently have attracted attention of quantum logicians. As expected from the short introduction, only commutative BCK- algebras (in particular, those having the relative cancellation property) are considered. Equivalence of bounded BCK-algebras with a special class of Abelian \(l\)-groups is demonstrated. By the way, bounded commutative BCK-algebras are equivalent to MV-algebras; on this ground, a new proof (using other methods) is obtained for the celebrated Mundici representation theorem for MV-algebras via unital Abelian \(l\)-groups. Chapter 6 gives an account of further algebraic properties of commutative as well as some other classes of BCK-algebras. Connections between various kinds of D-posets and BCK-algebras are presented. The last, seventh, chapter is devoted to Loomis-Sikorski style representation theorems for \(\sigma\)-complete MV-algebras and for BCK-algebras. These theorems lead to the calculus of observables for weakly divisible MV-algebras.
The book ends with a valuable 39 page bibliography.

MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
06D35 MV-algebras
06F35 BCK-algebras, BCI-algebras
06C15 Complemented lattices, orthocomplemented lattices and posets
03E72 Theory of fuzzy sets, etc.
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
03G25 Other algebras related to logic
81P68 Quantum computation
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