Is my understanding of "subshifts of finite type" correct? I am reading some lecture notes on Dynamical Systems, and I arrived at subshifts of finite type (ssft). The professor defines $\sum_n^+$ as the set of all one-sided sequences $.s_0s_1s_2...$ where for each $i$, $s_i \in \{0, 1, 2, ..., n-1\}.$ Then, let $A$ be any $n \times n$ matrix whose entries are in $\{0, 1\}$. Then the notes define the subshift of finite type corresponding to A as
$$\{\underline{s} = s_0 s_1 s_2...|\forall i: A_{s_i, s_{i+1}} = 1\}$$
I would like to show an example to make sure I understood the definition clearly. Let $A$ be the matrix
$$A = \begin{bmatrix}
1 & 1 & 0\\
0 & 1 & 1\\
1 & 1 & 0
\end{bmatrix}$$
I think we are supposed to label the rows as row 0, row 1, row 2, and the columns as col 0, col 1, col 2. N. We have
$$1 = A_{00} = A_{01} = A_{11} = A_{12} = A_{20} = A_{21}$$
$$0 = A_{02} = A_{10} = A_{22}$$
Therefore the sequences in $\sum_A^+$ are exactly those whose size-2 blocks are on this list: $00, 01, 11, 12, 20, 21$. So for example, an element of $\sum_A^+$ is
$$.00011120000000...$$
But an element not in $\sum_A^+$ is
$$.000020000.....$$
Is my understanding of ssft correct? Thank you so much!
 A: Yes, your understanding is correct. The set $\sum_n^+$ contains all one-sided infinite sequences over the alphabet $\{0, \ldots, n-1\}$. Sometimes it's useful to instead consider two-sided sequences: so the phrase "subshift of finite type", by itself, can be ambiguous. Your $\sum_A^+$ is a one-sided subshift of finite type. Also, when people say "subshift of finite type" they're usually talking about a slightly more complicated structure: not just the set of sequences, but also a particular topology on that set (namely, the one induced by the Tychonoff product topology on $\Sigma_n^+$) and a shift map $\sigma$, which slides a sequence to the left (or, in the one-sided case, deletes the first symbol: e.g., $\sigma(.121000\ldots) = .21000\ldots$).
The thing that makes it of "finite type" is that it can also be defined by a finite set of rules. An equivalent way of defining $\sum_A^+$ is to take all sequences that do not contain the forbidden words '02', '10', or '22'. If we wanted to, we could also forbid longer words like '01210'. The forbidden-words definition is more general: If we were to forbid longer words, there might not be a way of specifying the rule via a transition matrix. There is no matrix $A$ for which $\Sigma_A^+$ consists of all sequences that do not contain '01210'.
