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Partial residuated structures and quantum structures

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Abstract

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics—classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.

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Correspondence to Xiangnan Zhou.

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Project supported by the NSF of China (No. 10771524).

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Zhou, X., Li, Q. Partial residuated structures and quantum structures. Soft Comput 12, 1219–1227 (2008). https://doi.org/10.1007/s00500-008-0283-2

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