QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras

QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras

<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, as the common generalization of <...

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Main Authors: Yudan Du, Xiaohong Zhang
Format: Article
Language:English
Published: MDPI AG 2022-02-01
Series:Axioms
Subjects:
<i>BCI</i>-algebra
<i>BZ</i>-algebra
QM-<i>BZ</i>-algebra
quasi-hyper <i>BZ</i>-algebra
anti-grouped <i>BZ</i>-algebra
Online Access:https://www.mdpi.com/2075-1680/11/3/93
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author Yudan Du
Xiaohong Zhang
author_facet Yudan Du
Xiaohong Zhang
author_sort Yudan Du
collection DOAJ
description <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, as the common generalization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>C</mi></mrow></semantics></math></inline-formula>-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are proposed and their structures and constructions are studied. First, the definition of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is presented, and the structure of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is obtained: Each QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is KG-union of quasi-alter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>K</mi></mrow></semantics></math></inline-formula>-algebra and anti-grouped <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra. Second, the new concepts of generalized quasi-left alter (hyper) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebras and QM-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are introduced, and some characterizations of them are investigated. Third, the definition of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is proposed, and the relationships among <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra, and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are discussed. Finally, several special classes of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebras are studied in depth and the following important results are proved: (1) an anti-grouped quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is an anti-grouped <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra; (2) every generalized anti-grouped quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra corresponds to a semihypergroup.
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spelling doaj.art-ec63cbf5c8d84611b7fc4f07101a294b2023-11-24T00:28:42ZengMDPI AGAxioms2075-16802022-02-011139310.3390/axioms11030093QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-AlgebrasYudan Du0Xiaohong Zhang1School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an 710021, ChinaSchool of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an 710021, China<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, as the common generalization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>C</mi></mrow></semantics></math></inline-formula>-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are proposed and their structures and constructions are studied. First, the definition of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is presented, and the structure of QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is obtained: Each QM-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is KG-union of quasi-alter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>K</mi></mrow></semantics></math></inline-formula>-algebra and anti-grouped <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra. Second, the new concepts of generalized quasi-left alter (hyper) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebras and QM-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are introduced, and some characterizations of them are investigated. Third, the definition of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is proposed, and the relationships among <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra, quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>C</mi><mi>I</mi></mrow></semantics></math></inline-formula>-algebra, and quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra are discussed. Finally, several special classes of quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebras are studied in depth and the following important results are proved: (1) an anti-grouped quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra is an anti-grouped <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra; (2) every generalized anti-grouped quasi-hyper <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>Z</mi></mrow></semantics></math></inline-formula>-algebra corresponds to a semihypergroup.https://www.mdpi.com/2075-1680/11/3/93<i>BCI</i>-algebra<i>BZ</i>-algebraQM-<i>BZ</i>-algebraquasi-hyper <i>BZ</i>-algebraanti-grouped <i>BZ</i>-algebra
spellingShingle Yudan Du
Xiaohong Zhang
QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
Axioms
<i>BCI</i>-algebra
<i>BZ</i>-algebra
QM-<i>BZ</i>-algebra
quasi-hyper <i>BZ</i>-algebra
anti-grouped <i>BZ</i>-algebra
title QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
title_full QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
title_fullStr QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
title_full_unstemmed QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
title_short QM-<i>BZ</i>-Algebras and Quasi-Hyper <i>BZ</i>-Algebras
title_sort qm i bz i algebras and quasi hyper i bz i algebras
topic <i>BCI</i>-algebra
<i>BZ</i>-algebra
QM-<i>BZ</i>-algebra
quasi-hyper <i>BZ</i>-algebra
anti-grouped <i>BZ</i>-algebra
url https://www.mdpi.com/2075-1680/11/3/93
work_keys_str_mv AT yudandu qmibzialgebrasandquasihyperibzialgebras
AT xiaohongzhang qmibzialgebrasandquasihyperibzialgebras

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