Here we study the problem of constructing multivariate finite Markov chains whose coordinates are finite univariate Markov chains with given generator matrices. Specifically, we will be concerned here with construction of strong and weak Markov chain structures for a collection of finite Markov chains. We will use methods that are specific for Markov chains, and that are based on the results derived in Chapter 3. In this chapter we shall additionally be concerned with constructing weak Markov chain structures, which are related to the concept of weak Markov. Markov chain structures are key objects of interest in modeling structured dependence of Markovian type between stochastic dynamical given in terms of Markov chains. Accordingly, much of the discussion presented in this chapter is devoted to construction of Markov chain structures. Our construction allows for accommodating in a Markov structure model various dependence structures exhibited by phenomena one wants to model.