Provide an example of a universal algebra A that decomposes into a nontrivial direct product in which each factor is not isomorphic to the algebra A.

Question

A subalgebra A of a direct product $\prod_{k} A_{k}$ is a subdirect product of algebras $A_{k}$ if the projection A on each factor $A_{k}$ coincides with the factor itself. The subdirect product of algebras $\prod_{k} A_{k}$ is nontrivial if the projection of an algebra on any of the factors is not an isomorphism.

Question: Provide an example of a universal algebra A that decomposes into a nontrivial subdirect product in which each factor is not isomorphic to the algebra A (all product are isomorphic to A, but projection on each product is not bijection).

I have an example of such universal algebra: $C_{p}^{\infty} = \left \{z \in \mathbb{C} | z^{p^{k}} = 1 \right \}$ for some k, and some prime p. Where $\varphi: C_{p}^{\infty} \to C_{p}^{\infty}$ and $z \to z^{p}$ so $Ker \varphi = C_{p} \neq 1$

Answer 1

The intended problem might be this one:

Question: Provide an example of a universal algebra $A$ that decomposes into a nontrivial subdirect product in which each factor is not isomorphic to the algebra $A$ (all product factors are isomorphic to $A$, but the projection onto each product factor is not bijection).

If this is what is meant, then one could take $A$ to be any algebra satisfying $|A|>1$ that has an isomorphism $\iota\colon A\to A\times A$. In this situation, $\iota$ is a nontrivial subdirect embedding and $A$ is isomorphic to the factors in this subdirect product representation.

For a concrete choice of such an $A$, one could start with any nontrivial algebra $B$, let $A=B^{\omega}$, and let $\iota\colon B^{\omega}\to B^{\omega}\times B^{\omega}$ be defined by $\iota(b_0,b_1,b_2,\ldots) = ((b_0,b_2,\ldots),(b_1,b_3,\ldots)).$

(The proposed example of $C_p^{\infty}$ is subdirectly irreducible, so it does not have any nontrivial subdirect product representation.)