Mix $\star$-autonomous quantales and the continuous weak order

DOI10.1007/978-3-030-02149-8_12arXiv1807.06862MaRDI QIDQ6304353

Maria João Gouveia, Luigi Santocanale

Publication date: 18 July 2018

Abstract: 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 $Sigma=x,y,z,...$, 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 $d=mcard(Sigma)$. 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 $mathsf{L}_{vee}(mathbbI)$ of join-continuous functions from the unit interval to itself; the construction relies on a few algebraic properties of this quantale: it is $star$-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.

Mathematics Subject Classification ID

Quantales (06F07)

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