# The continuous weak order

## TL;DR

This paper extends the weak order lattice structure from permutations to continuous monotone functions on a cube, revealing new algebraic properties and embedding relationships with multinomial lattices.

## Contribution

It introduces a lattice structure on images of continuous monotone functions, generalizing discrete weak orders, and analyzes its algebraic and embedding properties.

## Key findings

- The lattice L(I^d) is self-dual and non-distributive.
- L(I^d) is generated by join-irreducible elements but has no completely join-irreducible elements.
- Embeddings of multinomial lattices into L(I^d) are characterized by subdivisions of the unit interval.

## 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 $\Sigma$ = {x,y,z,...}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card($\Sigma$) = 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($\Sigma$).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(I^d). 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 $\star$-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(I^d). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(I^d) 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(I^d) is only a join of meets of elements from the colimit of these embeddings.