Mix -autonomous quantales and the continuous weak order
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The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The 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, in dimension 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, where d = card(Σ). We show in this paper how to extend this order to images of continuous monotone paths from the unit interval to a d-dimensional cube. The key tool used to realize this construction is the quantale L_∨(I) of join-continuous functions from the unit interval to itself; the construction relies on a few algebraic properties of this quantale: it is -autonomous and it satisfies the mix rule. We begin developing a structural theory of these lattices by characterizing join-irreducible elements, and by proving these lattices are generated from their join-irreducible elements under infinite joins.
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