International Journal of Computational Intelligence Systems

Volume 12, Issue 1, November 2018, Pages 359 - 366

Further Complete Solutions to Four Open Problems on Filter of Logical Algebras

Authors
Wei Wang1, 2, 3, *, Pengxi Yang1, Yang Xu2, 3
1College of Sciences, Xi’an Shiyou University, Xi’an, Shaanxi, 710065, China
2National-Local Joint Engineering Laboratory of System Credibility Automatic Verification, Southwest Jiaotong University, Chengdu, 610031, China
3System Credibility Automatic Verification Engineering Lab of Sichuan Province, Southwest Jiaotong University, Chengdu, 610031, China
*Corresponding author. Email: wwmath@xsyu.edu.cn
Corresponding Author
Wei Wang
Received 3 January 2019, Revised 11 January 2019, Accepted 18 January 2019, Available Online 4 February 2019.
DOI
https://doi.org/10.2991/ijcis.2019.125905652How to use a DOI?
Keywords
Pseudo BCK-algebra; BL-algebra; Artificial intelligence; Filter; Boolean filter; Implicative filter; Normal filter; Fantastic filter
Abstract

This paper focuses on the investigation of filters of pseudo BCK-algebra and BL-algebra, important and popular generic commutative and non-commutative logical algebras. By characterizing Boolean filter and implicative filter in pseudo BCK-algebra, the essential equivalent relation between these two filters is revealed. An open problem that “In pseudo BCK-algebra or bounded pseudo BCK-algebra, is the notion of implicative pseudo-filter equivalent to the notion of Boolean filter?” is solved. Based on this, this paper explores the essential relations between the implicative (Boolean) filter and implicative pseudo BCK-algebra. A complete solution to an open problem that “Prove or negate that pseudo BCK-algebras is implicative BCK-algebras if and only if every filter of them is implicative filters (or Boolean filter)” is derived. This paper further characterizes the fantastic filter and normal filter in BL-algebra, then gets the equivalent relation between the two filters, and completely solves two open problems regarding the relationship between these two filters: 1. Under what suitable condition a normal filter becomes a fantastic filter? and 2. (Extension property for a normal filter) Under what suitable condition extension property for normal filter holds?

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
12 - 1
Pages
359 - 366
Publication Date
2019/02/04
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
https://doi.org/10.2991/ijcis.2019.125905652How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Wei Wang
AU  - Pengxi Yang
AU  - Yang Xu
PY  - 2019
DA  - 2019/02/04
TI  - Further Complete Solutions to Four Open Problems on Filter of Logical Algebras
JO  - International Journal of Computational Intelligence Systems
SP  - 359
EP  - 366
VL  - 12
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.2019.125905652
DO  - https://doi.org/10.2991/ijcis.2019.125905652
ID  - Wang2019
ER  -