= Lattice of convex lattice subgroups
The <convex lattice subgroups> of a <lattice-ordered group> form a distributive <lattice>, with intersection as <meet> and the convex lattice subgroup generated by a union as <join>. A positive element of a join is bounded by a finite product of positive elements from the generating subgroups. <Riesz decomposition in a lattice-ordered group> splits such an element into positive factors from those subgroups. If the original element belongs to another convex subgroup $E$, every factor does too. Consequently $E\cap(C\vee D)=(E\cap C)\vee(E\cap D)$.
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