Hyper BL-algebras

Abstract

We put forth the concept of hyper $BL$-algebras which is a generalization of  $BL$-algebras. We give some non-trivial examples and properties of hyper $BL$-algebras. Moreover, we introduce weak filters and weak deductive systems of hyper $BL$-algebras and study the relationships between them. Then we state and prove some theorems about weak filters and weak deductive systems. In particular, we define the concept of regular compatible congruence on hyper $BL$-algebras and   construct the quotient structure in hyper $BL-$algebras. Finally,  we discuss the conditions in which a quotient hyper $BL$-algebra is an $MV$-algebra.