Is there a name for the algebra (and its tensor products) given by generators $U_{j}$, $j \in \mathbb{Z}_{n}$ under the conditions $U_{j} = (1 - U_{j-1})(1-U_{j+1})$ ($j=1\implies j-1=n$ and $j=n\implies j+1=1$)?

There is no restriction on the commutativity of $U_{j}$. I am interested in structures for all possible cases for $U_{j}$.

When I say tensor product I think of generalization of following extension to multiple indices:

For two indices $$U_{j,j'} = \prod_{\substack{i,i'\in\{-1,0,1\}\\i\neq j\wedge i'\neq j}}(1 - U_{j+i,j'+i'})$$