Abstract
The relationships among lattice pseudoeffect algebras and some partial algebraic structures with residual implications are further studied, and the main results of the previous paper (titled residuation in lattice effect algebras) are extended comprehensively and systematically. First, we prove that every (commutative) quasiresiduated lattice is divisible, and every pseudoeffect algebra is good. Second, we prove some important results to show that effect (pseudoeffect) algebras and (commutative) quasiresiduated lattices can induce each other, and the induced relationships have the bidirectional reducing property. Finally, in order to explore the connections between pseudoeffect algebras (which may not be lattices) and residual partial algebras, we propose the concepts of Q-residuated partial monoids and Q-residuated lattices, and analyze deeply the induced relationships between pseudoeffect algebras and Q-residuated partial monoids. Moreover, we investigate the filters and congruence relations of (well) Q-residuated partial monoids and Q-residuated lattices.
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Acknowledgements
In the process of writing this article, we have obtained the help from Professors Yongjian Xie and Pengfei He at Shaanxi Normal University. We would like to express our heartfelt thanks to them!
Funding
This study was funded by the National Natural Science Foundation of China (No. 62081240416, 61976130) and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-698).
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Zhang, X., Wang, M. & Sheng, N. Q-residuated lattices and lattice pseudoeffect algebras. Soft Comput 26, 4519–4540 (2022). https://doi.org/10.1007/s00500-022-06839-w
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DOI: https://doi.org/10.1007/s00500-022-06839-w