[dasboth]

Yes that's right. The reason they're "Markov" chains is that they make the "Markov assumption" which is usually stated as "the future is independent of the past given the present". This just means that the probability of the next timestep depends only on what happened in the current one; it has no memory as such.

As detaro says:

"A Markov chain with memory for the last m steps is sometimes called a "Markov chain of order m""

So if you wanted it to depend on the last 5 timesteps, you'd create a 5th order Markov chain, and so on.