# Partial orders on $\mathbb{N}^m$ compatible with addition

I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $$\mathbb{N}^{m}$$ that are compatible with addition. That is, partial orders $$\leq$$ satisfying two additional axioms: $$\forall A \in \mathbb{N}^{m} \;\; (0, ..., 0) \leq A,$$ $$\forall A, B, C, D (A \leq B \text{ and } C \leq D) \rightarrow (A + C \leq B + D).$$

Apart from lexicographical order there exist many examples defined in terms of scalar products i.e. fix collection of vectors $$v_1, ..., v_k \in \mathbb{R}_{\geq 0}^{m}$$ and declare $$A \leq B$$ iff $$(B - A, v_j) \geq 0 \; \forall j$$.

• en.wikipedia.org/wiki/Monomial_order is very close to what you want at least; that might be a good place to start looking. May 11, 2021 at 19:34
• @SamHopkins Thanks. Indeed, that seems very close, but i'm not sure about the relation between the two. May 11, 2021 at 19:44
• You can also just use the pointwise ordering and many many more May 12, 2021 at 0:21
• @Sam, it seems equivalent to monomial order (with the $+$ operation). If the order is compatible with addition, and $A \le B$, then taking $D=C$ gives $A+C \le B+C$, so it is monomial. If the order is monomial, and $A\le B$ and $C \le D$, then $A+C \le B+C \le B+D$, where the first inequality is by adding $C$ to $A \le B$, and the second is by adding $B$ to $C \le D$. Jan 26 at 16:54