Prof. Nick Galatos, from the University of Denver, will give a seminar on the 9\(^{th}\) of December at 17:00, in the Aula Riunioni of the Department of Mathematics. The title is: *Densification of commutative residuated chains* (joint work with R. Horcik).

**Abstract**: It is well known that every countable totally-ordered set can be embedded into a countable totally-ordered and dense one (namely into the rationals, up to isomorphism). We prove that the same can be done for totally-ordered commutative monoids. Here commutativity is essential, as the result fails in the non-commutative setting. Using Zorn’s lemma it follows that it is enough to be able to perform one-step linear extensions namely to embed a totally-ordered commutative monoid with a gap into one that, even though not dense, at least resolves that particular gap.

We first embed the totally-ordered commutative monoind into a complete one, which then naturally carries the structure of a commutative residuated lattice, and prove that one-step linear extensions are possible in that richer setting, making use of the presence of an implication operation. The proof is inspired by the proof-theoretic elimination of the density rule in an appropriate sequent calculus in the setting of substructural logics, but none of this is visible in our presentation, which is purely algebraic.

In particular, we pass to the (semi)ring of polynomials (under join and multiplication) and try to loosely mimic the idea of the construction of field extensions. It turns out that the polynomials also form a module over that semiring, where the operation is meet and the action is implication. The task then is to identify a submodule that is totally-ordered, into which the original algebra embeds, and which contains, among others, an additional element that resolves the gap (and thus plays the role of a solution of a polynomial equation). The densifiability result was already known and our presentation builds on both the proof-theoretic and the algebraic understanding that the existing proofs provide.