Quantum B-algebras

• Autorzy: Wolfgang Rump
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Quantale; Non-commutative logic; Partially ordered group

Kategorie tematyczne

• 06F35: BCK-algebras, BCI-algebras
• 03G25: Other algebras related to logic
• 03F52: Linear logic and other substructural logics
• 03G27: Abstract algebraic logic
• 06F07: Quantales
• 06F15: Ordered groups
• 03G12: Quantum logic

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 11
• Strony: 1881-1899

Daty

wydano

2013-11-01

online

2013-08-23

Twórcy

autor

Wolfgang Rump

• University of Stuttgart

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-013-0302-0