In this paper we introduce a class of binary systems (groupoids,
algebras) $(X;*)$ on a set $X$ for which $x*y = (x*z)*(y*z)$ holds
for all  $x, y, z\in X$. These duplicative algebras include
surprisingly large classes of examples, including the $B$-algebras
which are closely related to groups as well as the left zero
semigroup. In order to study the structure theory of such algebras,
we introduce a graph $\Gamma_D(X)$ whose components have right ideal
characteristics and which determine certain related subalgebras
which are $B$-algebras.