Infimum of Subset Product in Ordered Group

Theorem
Let $\struct {G, \circ, \preceq}$ be an ordered group.

Let subsets $A$ and $B$ of $G$ admit infima in $G$.

Then:
 * $\map \inf {A \circ_\PP B} = \inf A \circ \inf B$

where $\circ_\PP$ denotes subset product.

Proof
This follows from Supremum of Product and the Duality Principle.

Also see

 * Supremum of Product