Let $X,\succ$ be a countable multiplicative abelian monoid with a partial order that is not a wellorder but is bounded below by $0$.
Let $Q\subsetneq X$ be a submonoid containing all its own multiplicative inverses.
Let $X\mapsto qX$ be an order isomorphism on $X$ for all $q\in Q$
This induces a monoidal quotient $X/Q$.
But does it induce some form of order-quotient ?
- $X,\succ/q$ for some $q\in Q$?
- $X,\succ/Q$?
In particular, what are sufficient prerequisites for the $X,\succ/Q$ to be a wellorder?
Example
I see this as analogous to the conditions for a quotient of a group to be torsion.
I have the example of $Q$ being a dense everywhere subset (multiplicative subgroup) of $X$ and $X$ being a dense everywhere subset (mutliplicative monoid) of $\Bbb Q^+$.
$\succ$ order-embeds in $>$ the standard order on $\Bbb Q^+$.
In this case, Zorn's lemma holds in $X,\succ$ and I need to show that it holds in $X/Q,\succ$, and that $Q$ is the minimal element by all chains of $X/Q,\succ$ and that $X/Q,\succ$ is a wellorder.