In this paper, we introduce a new notion, called an $BH$-algebra, which is a generalization of $BCH/BCI/BCK$-algebras. We define the notions of ideals and boundedness in $BH$-algebras, and show that there is a maximal ideal in bounded $BH$-algebras. Furthermore, we establish construct the quotient $BH$-algebras via translation ideals and obtain the fundamental theorem of homomorphisms for $BH$-algebras as a consequence.