Some types of filters in BL algebras

Abstract

In this paper we introduce some types of filters in a BL algebra A, and we state and prove some theorems which determine the relationship between these notions and other filters of a BL algebra, and by some examples we show that these notions are different. Also we consider some relations between these filters and quotient algebras that are constructed via these filters. (2000): 06F35, 03G25. Mathematics Subject Classification

Very true operators on MTL-algebras

“…x/ 2 F . We denote by L=F the set of congruence classes and L=F becomes an MTL-algebra with the natural operations induced by those of L. Note that a filter F of L is prime iff L=F is a linearly ordered MTL-algebra ( [2,6,7]). At the end of this section, we review the known main results about representation theory of MTL-algebras, which is helpful for studying very true analogous representation theorem of MTL-algebras.…”

“…From a logic point of view, various filters have natural interpretation as various sets of provable formulas. Recently, the filters on MTL-algebras have been widely studied and some important results have been obtained [2,[6][7][8]. In particular, Esteva introduced the idea of filters and prime filters in MTL-algebras to prove the completeness and chain completeness of MTL [2].…”

“…In particular, Esteva introduced the idea of filters and prime filters in MTL-algebras to prove the completeness and chain completeness of MTL [2]. After then, the concepts of implicative, positive and fantastic filters were defined in MTL-algebras in [6]. In [7], Borzooei was the first to systematically study filter theory in MTL-algebras, in which the relations between kinds of filters were obtained and some of their characterizations were presented.…”

On open problems based on fuzzy filters of pseudo BCK-algebras

“…For example, based on filters and prime filters in BL-algebras, Hájek proved the completeness of Basic Logic BL [18]. Literatures [1,4,5,8,12,17,24,30], further studied filters of BL-algebras, lattice implication algebras, pseudo BLalgebras, pseudo effect-algebras, residuated lattices, triangle algebras and the corresponding algebraic structures.…”

Vague BCK/BCI-algebras

“…BCI-algebras are generalizations of BCK-algebras. Most of the algebras related to the t-norm based logic, such as M T L-algebras, BL-algebras [3,4], hoop, M V -algebras and Boolean algebras et al, are extensions of BCK-algebras.…”