I have some problem to understand the following:
Let $X=\left\{0,1,2\right\}$ and consider $X^{\mathbb{Z}^d}, d\geq 2$ as being the set of all function from $Z^d$ to $X$. So for $\eta\in X^{\mathbb{Z}^d}$ it is either $\eta(x_i)=0$, $\eta(x_i)=1$ or $\eta(x_i)=2$.
Each $x\in Z^d$ is supposed to have 2d neighbours (for $d=2$ this means the point above, the point below, the point to the left and the point to the right).
Now the following definition is given.
We call a set of distinct points $x_0,x_1,\ldots,x_n=x_0$ a defect for $\eta$, if $x_{i+1}$ is a neighbor of $x_i$ for each $i$ and $$ \frac{1}{3}\sum_{k=0}^{n-1}\left\{\eta(x_{k+1})-\eta(x_k)\right\}\neq 0, $$ where the summands are all chosen mod 3 to be -1, 0 or 1 and the sum is ordinary addition, not mod 3.
My problem is: I do not see why the summands, when mod 3, are 0, 1 or -1.
What, if $\eta(x_{k+1})=2, \eta(x_k)=0$? Then the summand is 2 mod 3 =2....
It's telling you that for this purpose $2 - 0 = 2$ is replaced by $-1$ (since $2 \equiv -1 \mod 3$), and similarly $0 - 2 = -2$ is replaced by $1$. As to "why", I can't really say, because you haven't shown us what's going to be done with this definition.