Fantastic Ideals in BCI-Algebras

Processing of the certain information, especially inferences based on certain information. Is based on classical two-valued logic. Due to strict and complete logical foundation (classical logic), making inference levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of twovalued logic. Various kinds of non-classical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty. As it is well known, BCK and BCI-algebras are two classes of algebras of logic. They were introduced by Imai and Iseki [3, 4] and have been extensively investigated by many researchers. BCI-algebras are generalizations of BCK-algebras. Most of the algebras related to the t-norm based logic, such as MTLalgebras, BL-algebras, hoop, MV-algebras and Boolean algebras et al. [1] are extensions of BCK-algebras. This shows that BCK/BCI-algebras are considerably general structures. In this paper, the notion of fantastic ideal is introduced and some properties are established. By some examples we show that the relation of this notion and other types of ideals of BCI-algebra.