Monadic classes of quantum B-algebras
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Monadic classes of quantum B-algebras Lavinia Corina Ciungu1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The aim of this paper is to define the monadic quantum B-algebras and to investigate their properties. If the monadic operators are isotone, we show that they form a residuated pair. Special properties are studied for the particular case of monadic quantum B-algebras with pseudo-product, and a representation theorem for monadic quantum B-algebras with pseudo-product is proved. The monadic filters of monadic quantum B-algebras are defined, and their properties are studied. We prove that there is an isomorphism between the lattice of all filters of a monadic quantum B-algebra and the lattice of all filters of its subalgebra of fixed elements. Monadic operators on unital quantales are introduced, and the functional monadic quantale is constructed. Keywords Quantum B-algebra · Monadic quantum B-algebra · Monadic quantum B(pP)-algebra · Monadic filter · Monadic quantale · Relatively complete subalgebra · m-relatively complete subalgebra · Functional monadic quantale
1 Introduction The axiomatization of probability was done by A.N. Kolmogorov in 1933, and both probability and statistics had developed into major fields. But new areas of science have appeared during the last century, such as quantum mechanics, which do not satisfy the Kolmogorov axioms. In 1936, G. Birkhoff and J. von Neumann showed that the principles of quantum mechanics have different algebraic properties from a Boolean algebra, and the use of classical probability theory is not sufficient. Quantum logic, as a basis of quantum probability, deals with events which can be observed separately but not simultaneously. The new fields of sciences require a probability theory based on non-classical logics, and a lot of studies have been dedicated to the development of quantum structures. As it was mentioned in the special issue of Information Sciences Mesiar et al. (2009) dedicated to quantum structures, quantum information science is a new field of science and technology, combining and drawing on the disciplines of physical science, mathematics, computer science, quantum computing, and engineering (Dvureˇcenskij and Kuková 2016). The main directions of Communicated by A. Di Nola.
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Lavinia Corina Ciungu [email protected]
research for commutative quantum structures have been presented by Dvureˇcenskij and Pulmannová in (2000). In the last decades, developing algebraic models for non-commutative multiple-valued logics became a central topic in the study of fuzzy systems. The non-commutative generalizations of MV-algebras called pseudo-MV-algebras were introduced by Georgescu and Iorgulescu in (2001) and independently by Rach˚unek (2002) under the name of generalized MV-algebras. Pseudo-effect algebras were defined and investigated in Dvureˇcenskij and Vetterlein (2001) and Dvureˇcenskij and Vetterlein (2001) by Dvureˇcenskij and Vetterlein as non-commutative generalizations of effect algebras. Pseudo-BL-al
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