The finite algebras $(A,*,+)$ that satisfy the identity $(x*y)+(y*z)=(x+y)*(y+z)$ are precisely the algebras such that the one-dimensional cellular automata produced by $*$ and $+$ are commutative cellular automata. To be clear the operations $*,+$ do not necessarily satisfy any associativity or any other well known laws. See the paper [Commuting Cellular Automata][1] for details on this identity and its relation to cellular automata (or see [my answer here][2] for less details). I want to know if the variety $V$ of algebras that satisfy $(x*y)+(y*z)=(x+y)*(y+z)$ is generated by the collection of all its finite algebras. It suffices to show that the free algebras in $V$ are subdirect products of finite algebras in $V$. If the finite algebras $(A,*,+)$ satisfy other kinds of identities not implies by $(x*y)+(y*z)=(x+y)*(y+z)$, then what is an explicit example of such an identity?  If the free algebras in $V$ are subdirect products of finite algebras in $V$, then what is a good explicit example of how to embed the free algebras in $V$ as subdirect products of finite algebras in $V$?

Part of the motivation for this question is that cellular automata are usually over a finite alphabet $A$, so I wonder if finite algebras that satisfy $(A,*,+)$ that satisfy the identity $(x*y)+(y*z)=(x+y)*(y+z)$ satisfy any other identities.

  [1]: http://www.santafe.edu/media/workingpapers/97-08-076.pdf
  [2]: https://mathoverflow.net/a/156514/22277