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2 fixed typos

Let $G=\mathbb Z^2$. Every invariant linear order on $\mathbb Z^2$ is either induced from the standard order on $\mathbb R$ (or its inverse) by a linear map of the form $$(x,y)\mapsto x+\alpha y : \mathbb Z^2\to\mathbb R$$ where $\alpha\in\mathbb R\setminus\mathbb Q$, or is a composition of the lexicographic order $$(x,y)>(x',y') \quad\iff\quad x>x' \text{ or } (x=x' \text{ and } y>y')$$ up to with a bijective linear transformation $\mathbb Z^2\to\mathbb Z^2$.

Indeed, the set of elements of $\mathbb Z^2$ that are greater or equal to zero is a semi-group $H$ such that $H\cap(-H)=\{0\}$. The closed convex hull of such $H$ must be a half plane. If this half-plane is bounded by an irrational line, we have the first type of a linear order. If this half-plane is bounded by a rational line, then we may assume that it is a coordinate line $\{x=0\}$ (up to a linear change of coordinates in $\mathbb Z^2$), then the order is the lexicographic one (up to a change $y\mapsto -y$ depending on whether $(0,1)$ is "positive" or "negative" in this ordering).

In both cases, there are infinitely many elements between $(0,0)$ and $(1,0)$, hence it is not a word order.

1

Let $G=\mathbb Z^2$. Every invariant linear order on $\mathbb Z^2$ is either induced from the standard order on $\mathbb R$ (or its inverse) by a linear map of the form $$(x,y)\mapsto x+\alpha y : \mathbb Z^2\to\mathbb R$$ where $\alpha\in\mathbb R\setminus\mathbb Q$, or is a composition of the lexicographic order $$(x,y)>(x',y') \quad\iff\quad x>x' \text{ or } (x=x' \text{ and } y>y')$$ up to a bijective linear transformation $\mathbb Z^2\to\mathbb Z^2$.

Indeed, the set of elements of $\mathbb Z^2$ that are greater or equal to zero is a semi-group $H$ such that $H\cap(-H)=\{0\}$. The closed convex hull of such $H$ must be a half plane. If this half-plane is bounded by an irrational line, we have the first type of a linear order. If this half-plane is bounded by a rational line, then we may assume that it is a coordinate line $\{x=0\}$ (up to a linear change of coordinates in $\mathbb Z^2$), then the order is the lexicographic one (up to a change $y\mapsto -y$ depending on whether $(0,1)$ is "positive" or "negative" in this ordering).

In both cases, there are infinitely many elements between $(0,0)$ and $(1,0)$, hence it is not a word order.