In this paper we introduce the notion of a $BM$-algebra 
which is a specialization of $B$-algebras.
We show that the class of $BM$-algebras is a proper subclass of $B$-algebras 
and show that a  $BM$-algebra is  equivalent to  a  0-commutative $B$-algebra.
Moreover, we prove that a class of Coxeter algebras is a proper subclass 
of $BM$-algebras.