Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras
The concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough character...
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doaj-e915a83f3d054eeaaa390703a81ebec32020-11-25T03:16:57ZengMDPI AGMathematics2227-73902020-09-0181513151310.3390/math8091513Filters in Strong BI-Algebras and Residuated Pseudo-SBI-AlgebrasXiaohong Zhang0Xiangyu Ma1Xuejiao Wang2Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, ChinaDepartment of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, ChinaDepartment of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, ChinaThe concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough characteristics to describe residual implications in-depth, so we propose a new concept of strong BI-algebra, which is exactly the algebraic abstraction of fuzzy implication with pseudo-exchange principle (PEP) property. Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebras to the non-commutative case, and propose the concept of pseudo-strong BI (SBI)-algebra, which is the common extension of quantum B-algebras, pseudo-BCK/BCI-algebras and other algebraic structures. We establish the filter theory and quotient structure of pseudo-SBI- algebras. Moreover, based on prequantales, semi-uninorms, t-norms and their residual implications, we introduce the concept of residual pseudo-SBI-algebra, which is a common extension of (non-commutative) residual lattices, non-associative residual lattices, and also a special kind of residual partially-ordered groupoids. Finally, we investigate the filters and quotient algebraic structures of residual pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems.https://www.mdpi.com/2227-7390/8/9/1513basic implication algebra (BI-algebra)strong BI-algebrapseudo-SBI-algebrafilterresidual pseudo-SBI-algebra |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Xiaohong Zhang Xiangyu Ma Xuejiao Wang |
spellingShingle |
Xiaohong Zhang Xiangyu Ma Xuejiao Wang Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras Mathematics basic implication algebra (BI-algebra) strong BI-algebra pseudo-SBI-algebra filter residual pseudo-SBI-algebra |
author_facet |
Xiaohong Zhang Xiangyu Ma Xuejiao Wang |
author_sort |
Xiaohong Zhang |
title |
Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras |
title_short |
Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras |
title_full |
Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras |
title_fullStr |
Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras |
title_full_unstemmed |
Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras |
title_sort |
filters in strong bi-algebras and residuated pseudo-sbi-algebras |
publisher |
MDPI AG |
series |
Mathematics |
issn |
2227-7390 |
publishDate |
2020-09-01 |
description |
The concept of basic implication algebra (BI-algebra) has been proposed to describe general non-classical implicative logics (such as associative or non-associative fuzzy logic, commutative or non-commutative fuzzy logic, quantum logic). However, this algebra structure does not have enough characteristics to describe residual implications in-depth, so we propose a new concept of strong BI-algebra, which is exactly the algebraic abstraction of fuzzy implication with pseudo-exchange principle (PEP) property. Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebras to the non-commutative case, and propose the concept of pseudo-strong BI (SBI)-algebra, which is the common extension of quantum B-algebras, pseudo-BCK/BCI-algebras and other algebraic structures. We establish the filter theory and quotient structure of pseudo-SBI- algebras. Moreover, based on prequantales, semi-uninorms, t-norms and their residual implications, we introduce the concept of residual pseudo-SBI-algebra, which is a common extension of (non-commutative) residual lattices, non-associative residual lattices, and also a special kind of residual partially-ordered groupoids. Finally, we investigate the filters and quotient algebraic structures of residual pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems. |
topic |
basic implication algebra (BI-algebra) strong BI-algebra pseudo-SBI-algebra filter residual pseudo-SBI-algebra |
url |
https://www.mdpi.com/2227-7390/8/9/1513 |
work_keys_str_mv |
AT xiaohongzhang filtersinstrongbialgebrasandresiduatedpseudosbialgebras AT xiangyuma filtersinstrongbialgebrasandresiduatedpseudosbialgebras AT xuejiaowang filtersinstrongbialgebrasandresiduatedpseudosbialgebras |
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1724633984971833344 |