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

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?

Computational Intelligence for Emerging Systems and Applications

“…The first paper in this special issue [17] comes back to main theoretical backgrounds of fuzzy logic, which represents a natural basis of many intelligent methodologies, solving an open problem relative to the behavior of Boolean filters within algebraic structures.…”

“…We hope readers will enjoy this special issue and that its reading will help in their research and the implementation of specific tools and applications of computational intelligence, offering an interesting approach to key issues in computational intelligence, from basic mathematical issues [17] and data structures [18], to the management of emotions [24] and decision-making [25], addressing intermediate knowledge processing tools like concept representation [19], visualization of information [20] and its evaluation [21], plus software analysis [22] and spam detection [23]. Yun Li, Tianrui Li, and Javier Montero, editors.…”