Abstract
Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) arising from quantum computational logics are the generalizations of both quasi-MV algebras and pseudo-MV algebras. In this paper, we introduce the notions of states, state-morphisms, state operators and state-morphism operators to quasi-pMV algebras. First, we present the related properties of states on quasi-pMV algebras and show that states and Bosbach states coincide on any quasi-pMV algebra. And then we investigate the relationship between state-morphisms and the normal and maximal ideals of quasi-pMV algebras. We prove state-morphisms and extremal states are equivalent. The existence of states on quasi-pMV algebras is also discussed. Finally, state operators and state-morphism operators are introduced to quasi-pMV algebras, and the corresponding structures are called state quasi-pMV algebras and state-morphism quasi-pMV algebras, respectively. We investigate the related properties of ideals under state operators and state-morphism operators. Meanwhile, we show that there is a bijective correspondence between normal \(\sigma \)-ideals and ideal congruences on state quasi-pMV algebras.
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A word of caution is necessary here. We define “locally finite" as the similar notion of MV-algebras. However, in the usual algebraic meaning, “locally finite" is used as “finitely generated subalgebras are finite".
In a quasi-pMV algebra, if \(0=1\), then it is called flat.
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Acknowledgements
This study was funded by the National Natural Science Foundation of China (Grant No. 11501245), China Postdoctoral Science Foundation (No. 2017M622177) and Shandong Province Postdoctoral Innovation Projects of Special Funds (No. 201702005).
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Chen, W., Dudek, W.A. States, state operators and quasi-pseudo-MV algebras. Soft Comput 22, 8025–8040 (2018). https://doi.org/10.1007/s00500-018-3069-1
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DOI: https://doi.org/10.1007/s00500-018-3069-1