We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

翻译：我们证明了配备分量序的$q$-递减词偏序集构成一个格。对于任意$q>0$，我们枚举了其并不可约元素；对于任意正有理数$q$，我们确定了覆盖数、区间数以及交不可约元素的数量。后者呈现出与在$2\lceil q\rceil+1$个字母的字母表上避免$\lceil q\rceil^2+2\lceil q\rceil-1$个长度为2的连续模式的词相同的结构。此外，我们分析了这些数量中若干项的渐近行为。