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Pseudo-involutive residuated lattices (non-commutative) and pseudo-effect algebras. (English) Zbl 1334.06010
Summary: The notion of pseudo-involutive residuated lattices (non-commutative) is introduced. By introducing two partial operations in pseudo-effect algebras, the mutual relationship between pseudo-involutive residuated lattices and lattice pseudo-effect algebras is investigated. The following results are proved: a lattice pseudo-effect algebra under certain conditions can be extended to a pseudo-involutive residuated lattice and the latter with certain properties can be restricted to the former. Especially, a sufficient and necessary condition for a pseudo-involutive residuated lattice to be lattice pseudo-effect algebra with the Riesz decomposition property is obtained. Finally, the ideals and filters of pseudo-effect algebras and pseudo residuated lattices are investigated.
MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
03G12 Quantum logic
06D35 MV-algebras
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