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Quantum B-algebras. (English) Zbl 1326.03077

Summary: The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale \(Q\) is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of \(Q\). The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

MSC:

03G12 Quantum logic
03F52 Proof-theoretic aspects of linear logic and other substructural logics
03G25 Other algebras related to logic
06F07 Quantales
06F15 Ordered groups
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