Logarithmic order of free distributive lattice
Koichi I. Yamamoto · Journal of the Mathematical Society of Japan · 1954
l.-Introduction.-Theproblem to determine the order $f(n)$ of the free distributive lattice $FD(n)$ generated by $n$ symbols $\gamma_{1},$ $\cdots,$ $\gamma_{n}$ was first proposed by Dedekind, but very little is known about this number [1, p. 146].Only the first six values of $f(n)$ are computed, and enumerations of further $f(n)$ appear to lie beyond the scope of any reasonable methods known today.It might, however, be pointed out that Morgan Ward, who found $f(6)$ by the help of computing machines, stated [2] an asymptotic relation $\log_{2}\log_{2}f(n)\sim n$and that the present author proved in a previous note [3] thatThe author cannot prove or disprove this interesting relation, but he proves in the present paper that $\sqrt{\frac{2}{\pi}}n-1-1_{-\log_{2}\sqrt{\frac{n\pi}{2}}(1+O(n^{-I}))}$ (Theorem 2), which in particular implies that for an $arbitr_{/}ary$ positive constant $\delta$ $2^{\pi}n^{-1}z^{-\delta}<\log_{2}f(n)<2^{n}n^{-1}\tau^{+\delta}$ if $n$ is sufficiently large, and that