Abstract
In this paper, we introduce quasi-MV* algebras as the generalization of MV*-algebras and quasi-MV algebras. First, we give the definition of a quasi-MV* algebra and investigate the related properties of a quasi-MV* algebra. We also discuss the relationship between quasi-MV* algebras and quasi-MV algebras. Second, we introduce the notions of ideals and weak ideals in a quasi-MV* algebra. The basic properties of ideals are presented, and the relationship between ideals and ideal congruences on a quasi-MV* algebra is discussed. Finally, we study the filters of a quasi-MV* algebra. We give the equivalent characterization of a filter and show that there exists a bijective correspondence between filters and ideals in a quasi-MV* algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Bou F, Paoli F, Ledda A, Giuntini R (2010) The logic of quasi-MV algebras. J Log Comput 20:619–643. https://doi.org/10.1093/logcom/exp080
Bou F, Paoli F, Ledda A, Freytes H (2008) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part II. Soft Comput 12:341–352. https://doi.org/10.1007/s00500-007-0185-8
Chang CC (1958) Algebraic analysis of many-valued logics. Trans Am Math Soc 88:467–490. https://doi.org/10.2307/1993227
Chang CC (1963) A logic with positive and negative truth values. Acta Philos Fenn 16:19–39
Jipsen P, Ledda A, Paoli F (2013) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part IV. Rep Math Logic 48:3–36. https://doi.org/10.4467/20842589RM.13.001.1201
Kowalski T, Paoli F (2010) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part III. Rep Math Logic 45:161–199
Ledda A, Konig M, Paoli F (2006) MV algebras and quantum computation. Stud Log 82:245–270. https://doi.org/10.1007/s11225-006-7202-2
Lewin RA, Sagastume M (2002) Paraconsistency in Chang’s logic with positive and negative truth values. In: Carnielli WA, Coniglio ME, D’Ottaviano IML (eds) Paraconsistency, the Logical Way to the Inconsistent. Marcel Dekker, New York, pp 381–396
Lewin R, Sagastume M, Massey P (2004) MV*-algebras. Log J IGPL 12:461–483. https://doi.org/10.1093/jigpal/12.6.461
Lewin R, Sagastume M, Massey P (2004) Chang’s \({{{{L}}}^{*}}\) Logic. Log J IGPL 12:485–497. https://doi.org/10.1093/jigpal/12.6.485
Paoli F, Ledda A, Giuntini R, Freytes H (2009) On some properties of quasi-MV algebras and \({\sqrt{^{\prime }}}\)quasi-MV algebras. Part I. Rep Math Logic 44:31–63
Funding
This work was supported by Shandong Provincial Natural Science Foundation, China (No. ZR2020MA041), China Postdoctoral Science Foundation (No. 2017M622177) and Shandong Province Postdoctoral Innovation Projects of Special Funds (No. 201702005).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Author A declares that she has no conflict of interest. Author B declares that she has no conflict of interest.
Human and animal participants
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Jiang, Y., Chen, W. Quasi-MV* algebras: a generalization of MV*-algebras. Soft Comput 26, 6999–7015 (2022). https://doi.org/10.1007/s00500-022-07223-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-022-07223-4