Mix *quantales and the continuous weak order
03/10/2018 FCUL, 16h30, 6.2.33
Luigi Santocanale (AixMarseille Université)
Faculty of Sciences, University of Lisbon
The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order.
This lattice structure is generalized to the set of words on a fixed alphabet Σ = { x, y, z, . . . }, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card(Σ) = 2, as lattices of lattice paths. By interpreting the letters x, y, z, . . .as axes, these words can be interpreted as discrete increasing paths on a grid of a ddimensional cube, with d = card(Σ).
In this talk I’ll explain how to extend this order to images of continuous monotone functions from the unit interval to a ddimensional cube. The lattice so obtained is denoted L(I^d). The key tool used to realize this construction is the quantale Q∨(I) of joincontinuous functions from the unit interval to itself; the construction relies on a few algebraic properties of this quantale: it is involutive (that is, cyclic, non commutiative and *autonomous, often called a Girard quantale since it is a model of classical linear logic) and it satisfies the mix rule.
We begin developing a structural theory of the lattices L(I^d): they are selfdual, they are generated under infinite joins from their joinirreducible elements, they have no completely irreducible elements nor compact elements.
The colimit of all the ddimensional multinomial lattices embeds into L(I^d) by taking rational coordinates.
When d = 2, L(I^d) = Q∨(I) is the DedekindMacNeille completion of this colimit. When d ≥ 3, every element of L(I^d) is a join of meets of elements from this colimit.
