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Open AccessFeature PaperArticle

Q-Filters of Quantum B-Algebras and Basic Implication Algebras

1
Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China
2
Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
3
Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran
4
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(11), 573; https://doi.org/10.3390/sym10110573
Received: 26 September 2018 / Revised: 25 October 2018 / Accepted: 26 October 2018 / Published: 1 November 2018
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
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PDF [298 KB, uploaded 19 November 2018]

Abstract

The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established. View Full-Text
Keywords: fuzzy implication; quantum B-algebra; q-filter; quotient algebra; basic implication algebra fuzzy implication; quantum B-algebra; q-filter; quotient algebra; basic implication algebra
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Zhang, X.; Borzooei, R.A.; Jun, Y.B. Q-Filters of Quantum B-Algebras and Basic Implication Algebras. Symmetry 2018, 10, 573.

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