Suppose that you have a finite state Markov chain, with $n$ states and characterized by $p_{i,j}$ the probability of reaching state $j$ from state $i$. Consider the new Markov chain with $n^2$ states and transition probabilities $$p_{i_1i_2,j_1j_2}=p_{i_2,j_1}p_{j_1,j_2}.$$ Note that, for all $i_a$ and $i_b$, $$p_{i_ai_2,j_1j_2}=p_{i_bi_2,j_1j_2}.$$
Now the idea behind this "product" is to study the periodicity of the new chain. For some trivial examples this transform makes a periodic chain become aperiodic; think of a chain with two states with $p_{1,0}=p_{0,1}=1$. Then my question is:
Does the iterated application of this "product" render any chain aperiodic?
I guess there should be some easy counterexample but so far didn't come up with any.