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.

中文翻译：

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连续弱序

有限集上的置换集可以给出称为弱布鲁哈阶的晶格结构。这种格结构被推广到固定字母表 $\Sigma$ = {x,y,z,...} 上的一组单词，其中每个字母都有固定的出现次数。这些格称为多项格，当 card($\Sigma$) = 2 时，称为格路径的格。通过解释字母 x, y, z, . . . 作为轴，这些词可以解释为 d 维立方体网格上的离散递增路径，d = card($\Sigma$)。我们展示了如何将这种排序扩展到单位间隔的连续单调函数的图像到一个 d 维立方体并证明这个排序是一个格子，用 L(I^d) 表示。这种构造依赖于从实数的单位区间到自身的连接连续函数的量子数的一些代数性质：它是循环 $\star$-自治的并且它满足混合规则。我们研究这些格子的结构特性，这是自我双重的而不是分配的。我们描述了连接不可约元素，并表明这些晶格是在无限连接下从它们的连接不可约元素生成的，它们没有完全连接不可约元素也没有紧凑元素。然后我们研究将 d 维多项式点阵嵌入到 L(I^d) 中。我们表明这些嵌入是从单位间隔的细分函数中产生的，并观察到 ​​L(I^d) 是这些嵌入的共限的 Dedekind-MacNeille 完成。然而，