  We study two variations of the Brauer algebra $B_n(x)$. The first is the
algebra $A_n(x)$, which generalizes the Brauer algebra by considering loops.
The second is the algebra $L_n(x)$, the $A_n(x)$-subalgebra generated by
diagrams without horizontal arcs. $A_n(x)$ and $L_n(x)$ have for $x \neq 0$ an
hereditary-chain indexed by all integers. Following the ideas of Martin in the
context of the partition algebra, and Doran et al. for the Brauer algebra, we
study semisimplicity of $A_n(x)$ using restriction and induction in $A_n(x)$
and $L_n(x)$. Our main result is that $A_n(x)$ is semisimple if $x \notin Z$
and that $L_n(x)$ is semisimple if $x \neq 0$.
