Further Complete Solutions to Four Open Problems on Filter of Logical Algebras
- https://doi.org/10.2991/ijcis.2019.125905652How to use a DOI?
- Pseudo BCK-algebra; BL-algebra; Artificial intelligence; Filter; Boolean filter; Implicative filter; Normal filter; Fantastic filter
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?
- © 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 -