Consider the following potential on three nodes.
represented by the following factor graph.
Now the notes claim that we can represent this factor graph as both a Bayesian network and a Markov networks as follows.
The representation is given here 
I'm failing to see the connection with the extra three nodes $Z_1,Z_2,Z_3$? What would the break down into clique potentials of the probability distribution looks like? Since there are only 2-cliques, the theory as i know it suggests that $p(x_1,x_2,x_3,Z_1,Z_2,Z_3) = 1/Z \psi_1(x_1,Z_1) \psi_2(Z_1,x_2) \psi_3(x_2,Z_2) \psi_4(Z_2,x_3) \psi_5(x_3,Z_3) \psi_6(Z_3,x_1)$ where $Z$ is the normalising constant. However i'm confused as to what $\psi(Z_1)$ would represent and how it equals $f_a(x_1,x_2)$? Can anyone fill in the missing pieces of the construction?