In this paper we show that the class of $PC$-algebras and the class
of $B$-algebras with condition $(D)$ are Smarandache disjoint, and
show that an algebra $(X;*, 0)$ is a Coxeter algebra if and only
if it is a $PC$-algebra with $(N)$. Moreover, we show that there
is no non-trivial quadratic $PC$-algebras on a field with
$|X|\geq3$.