The continuous weak order

Abstract

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 d-dimensional cube, with d=card(Σ).We show how to extend this ordering to images of continuous monotone functions from the unit interval to a d-dimensional cube and prove that this ordering is a lattice, denoted by L(Id). This construction relies on a few algebraic properties of the quantale of join-continuous functions from the unit interval of the reals to itself: it is cyclic ⋆-autonomous and it satisfies the mix rule.We investigate structural properties of these lattices, which are self-dual and not distributive. We characterize join-irreducible elements and show that these lattices are generated under infinite joins from their join-irreducible elements, they have no completely join-irreducible elements nor compact elements. We study then embeddings of the d-dimensional multinomial lattices into L(Id). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(Id) is the Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we restrict to embeddings that take rational values and if d>2, then every element of L(Id) is only a join of meets of elements from the colimit of these embeddings.