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 Subset Product in Ordered Group and the Duality Principle.

$\blacksquare$

Also see

• Supremum of Subset Product in Ordered Group