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(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 ⋆-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.
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Contributor : Luigi Santocanale <>
Submitted on : Wednesday, December 5, 2018 - 4:44:36 PM
Last modification on : Tuesday, December 18, 2018 - 1:24:35 AM
Long-term archiving on : Wednesday, March 6, 2019 - 12:22:35 PM


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  • HAL Id : hal-01944759, version 1
  • ARXIV : 1812.02329


Luigi Santocanale, Maria João Gouveia. The continuous weak order. 2018. ⟨hal-01944759v1⟩



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