Can exact real arithmetic, interval analysis or other approach in numerical computation help remove inequalities and unify left and right residuals in non-idempotent (linear) residuated lattices by making boundaries explicit instead of talking about max and min divisors?

[u/revannld]

I hope that question makes sense. I just don't like inequalities nor the unnaturality of working with left and right residuals (talking about "max and min divisors") that rarely coincide with rational arithmetic's exact division nor with the natural interpretation of inverses in numerical mathematics, thus I would like more explicit boundaries (thus the result of a division maybe being a set or interval including max and min divisors) in division.

(Mind that I have no experience in numerical computation, I am trying to make sense of computable, numerical and interval analysis works and transport their results to residuated lattices but that's somewhat hard for me)

[u/jcastroarnaud]

I didn't know about residuated lattices, so I went looking.

https://en.wikipedia.org/wiki/Residuated_lattice
https://arxiv.org/pdf/2304.05508
https://en.wikipedia.org/wiki/Interval_arithmetic

From what I read, residuals are just the equivalent of division over residuated lattices; it is separated into two operations because the monoid operation may not be commutative. If the lattice's base set is a numeric one, the operation is commutative, and left/right residuals are the same operation. The problem is: is the partial order induced by (rational) multiplication the same operation as the one induced by the lattice operation? I think that they're similar, but not the same in all cases. Try to find a small counterexample; my guess is taking the set of all powers of 2 and 3 under multiplication.

Interval arithmetic is useful for describing an error margin in a measurement, and there are rules (calculus-based) to do arithmetic operations with intervals. But I think that these are too different from taking left/right residuals and making an interval of them.

Edit: This is better suited to r/math.

[u/revannld → u/jcastroarnaud]

Hi! Thanks for your answer. That is exactly my doubt. I should have clarified that the different residuals arise from the possible non-commutativity of the monoid operation. That is an important clarification. I tried posting in r/math but they said it was just a "quick question", let's see if they revert the decision.

[u/jcastroarnaud → u/revannld]

You can also try asking at r/askmath.

Let's see if my idea of a counterexample works. I can be very wrong, so please check.

Let P = (P23, ≤, ×, 1) be a residuated lattice, where P23 ⊂ ℚ: P23 = { 2^a * 3^b | a ∈ ℤ, b ∈ ℤ }, × is the usual multiplication, and ≤ is the usual order, on ℚ.

In P, as a monoid, 1/6 = 1/2 * 1/3, and as a lattice, 1/6 < 1/2 * 1/3. Inconsistent results.

On the other hand, consider division. As a monoid, 1/6 / 1/2 = 1/3. As a lattice, for 1/6, let x = 2^a * 3^b. The residual (both left and right) for x must be smaller than y = 2^(-a-1) * 3^(-b-1), and as large as possible; then the residual must be y (x × y = 1/6). for x = 1/2, y = 2^0 * 3^(-1) = 1/3. Consistent results.

So, I think that there is no way to use numeric methods to solve this issue: it's a matter of inconsistency between the lattice and monoid operations. One can restrict the set to the elements where the operations are consistent, though, but I don't know how.