A study on doubt fuzzy BCK/BCI-algebras and other algebraic structures

Abstract

Fuzzy set theories provide strict mathematical frameworks using vague phenomena and can be precisely and rigorously studied. The core theories in mathematics such as algebra, topology, lattices and analysis are subject to fuzzi cation. Many researchers have dealt with subalgebras, ideals, subimplicative ideals, P-ideals, H-ideals, implica- tive ideals, hyper structure etc. and their fuzzi cation. This study attempts to introduce a new class of algebraic structures in intuitionis- tic fuzzy set and interval-valued fuzzy set. The concepts of doubt intuitionistic fuzzy subalgebra/ideals, doubt intuitionistic fuzzy H-ideals, doubt intuitionistic fuzzy sub- implicative ideals, doubt intuitionistic fuzzy P-ideals and doubt intuitionistic fuzzy translations of doubt intuitionistic fuzzy sub-implicative ideals are introduced for BCK/BCI-algebra and their properties are duly characterized. The notion of direct product of two doubt intuitionistic fuzzy subalgebras and two doubt intuitionistic fuzzy H-ideals of two BCK/BCI-algebras has also been discussed and some related properties have investigated. We have also found that the direct product of two intuitionistic fuzzy sets becomes doubt intuitionistic fuzzy H-ideals and doubt intuitionistic fuzzy subalgebras if and only if for any t; s 2 [0; 1], upper and lower level sets are H-ideals or subalgebras of BCK/BCI algebra X Y . Results of doubt interval-valued fuzzy subalgebras and doubt interval-valued fuzzy ideals in BCK-algebras and BF-algebras are also provided. Cartesian product of two interval-valued fuzzy sets is de ned and its properties are established. The relation between doubt intuitionistic fuzzy ideals and an upper level and lower level cuts of BCK/BCI-algebra have also been explored. Besides doubt intuitionistic fuzzy hyper lters in hyper BE-algebras have also been established in the present work.