In any lattice ordered group, we say that a convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that $X\subseteq P$ or $Y\subseteq P$. This is analogous to the notion of a prime ideal, which in turn is analogous to the notion of a prime number in the integers.
Consider the following real-world l-group:
I have a shopping cart full of items. I can add or remove items from the cart. (Isomorphic to $(\mathbb Z^n,+)$ where $(1,2)$ inidicates 1 apple and two oranges or some such.) The meet of two carts is the componentwise minimum, e.g. $(5,2,3)\wedge(4,6,4)=(4,2,3)$ and similarly with join.
I'm having trouble visualizing the prime subgroups here. An alternative definition of prime is that $f\wedge g=1$ implies $f\in P$ or $g\in P$, so I think the prime subgroups are of the form $(0,x_1,x_2,\dots)$ i.e. the prime subgroup $P_i$ is generated by the element $(1,1,\dots,0,1,\dots)$ that is 1 everywhere except the ith position.
These subgroups don't appear to correspond to any intuitive "building blocks" of the group like I would expect. What am I not understanding?