ProofWiki:Sandbox

Definition
Let $\mathbf A = \sqbrk a_k$ be a logical matrix for a $k\geq 2$.

Let
 * $X _{\mathbf A} = \set {x=\sequence {x_n} _{n \in \Z} : x_n \in \set {1,2,\ldots, k}, a _{x_n, x_{n+1}} = 1}$

Let $\sigma _{\mathbf A} : X _{\mathbf A} \to X _{\mathbf A}$ be the forward shift operator, i.e.
 * $\sigma _{\mathbf A}(x) := y$

where $y_n = x_{n+1}$ for all $n\in\Z$.

Then the pair $\struct { X _{\mathbf A}, \sigma _{\mathbf A} }$ is called a shift of finite type.