To let $ R $ be a commutative ring. A saturated monoid in $ R $ is a multiplicative submonoid $ S subset R $ which is closed among dividers, i.e. $ xy in S implies x in S $. This is the reverse of the analog axiom for ideals $ x in I implies xy in I $. Saturated monoids and ideals thus have a kind of duality, of which part swaps addition and multiplication and part reverses the direction of the implication.

The only elements given to us suggest a pair of operations between ideals and saturated monoids

$$ begin {gathered} mathrm {Ideal} (R) rightleftarrows mathrm {SatMon} (R), \ I mapsto (1 + I) _ { text {sat}} text {and} S . mapsto left langle S-1 right rangle. end {gathered} $$

Take those from the powerset of induced poset structures $ R $, given by inclusions, both operations above are poset morphisms (take inclusions to inclusions). However, they are not additional functors:

$$ begin {assembled} left langle S-1 right rangle ⊂ I iff S-1 subset I iff S subset 1 + I \ S supset (1 + I) _ text { sat} iff S supset 1 + I end {gathered} $$

This is a strange situation for functors $ mathsf C rightleftarrows mathsf D $ say $ F, G $ and yet bijections $$ mathsf D (FA, B) cong mathsf C (GB, A) = mathsf C ^ text {op} (A, GB). $$

**Question 1.** Are the above operations based on a missing addition?

If not then:

**Question 2.** Is the formal duality between ideals and saturated monoids captured by another adjunction?

If not then:

**Question 3.** Is there more structure in the abstract environment I described above with interesting category theory?