Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

Xiaohong Zhang, Xiangyu Ma and Xuejiao Wang
Additional contact information
Xiaohong Zhang: Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China
Xiangyu Ma: Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China
Xuejiao Wang: Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, China

Mathematics, 2020, vol. 8, issue 9, 1-27

Abstract: 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). Furthermore, in order to describe the characteristics of the algebraic structure corresponding to the non-commutative fuzzy logics, we extend strong BI-algebra 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 residuated pseudo-SBI-algebras, and obtain a unity frame of filter theory for various algebraic systems.

Keywords: basic implication algebra (BI-algebra); strong BI-algebra; pseudo-SBI-algebra; filter; residuated pseudo-SBI-algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/9/1513/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/9/1513/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:9:p:1513-:d:409028

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().