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

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) 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.

A Class of BCI-Algebra and Quasi-Hyper BCI-Algebra

“…In general, this paper discusses the relationship between (hyper) logical algebra and classical abstract algebra, and the description of the structure of (hyper) logical algebra is more clear. As a further research topic, we can consider exploring the internal connections between (hyper) BCI-algebras, BI-algebras and CA-semihypergroups (see [36][37][38]).…”

On Filters of Bitonic Algebras

“…Filters, in general, provide accurate keys to locate elements that are large enough to perform some measures which are beneficial in general topology, firstly studied by Cartan in [2], in logic, studied by Tarski, Moisil and others, many of whose results are found in Birkhoff's Lattice Theory in [3]. Relying on several theory-based investigators who dealt with filters, such as Xiaohong, Xiangyu and Xuejiao, the connection and ranked notion of many filter concepts were studied in a systematic manner, and the structure they used is especially essential in their studies given in [4]. Lately, many researchers have focused on filters of various algebraic structures.…”