Definition of Jordan–Dedekind chain condition Let $(X,\le)$ be a lattice. Which is the correct definition for Jordan-Dedekind condition:
1) all maximal chains in $X$ have the same cardinality.
2) for any interval $I$ in $X$, all maximal chains in $I$ have the same cardinality.
Are these equivalent?
Definition (1) seems to be more frequent. But in an old paper Jordan–Dedekind chain condition is defined with (2) for a finite lattice.
 A: (1) does not imply (2). Consider $N_5$ lattice and add a countable chain $\omega$ to its top (the greatest element of $N_5$ and the least element of $\omega$ are glued together). Constructed lattice does not satisfy (2) because $N_5$ has two maximal chains of length $2$ and $3$ respectively. Nonetheless, this lattice has only two maximal chains and both of them are countable, so (1) holds.
(1) is called Jordan-Hölder condition in Grätzer's book, while
(2) is called Jordan-Dedekind chain condition in Birkhoff's book and on wiki.
$Updated:$ In case of finite lattice (2) obviously implies (1). Assume that (2) does not hold: there exist two maximal chains $C_1, C_2$ in some interval $I = [a, b]$ such that $|C_1| \neq |C_2|$. Both of them are contained in some maximal chains $M_1, M_2$, which can be choosen such that $M_1 \setminus I = M_2 \setminus I$, since we can construct a maximal chains in $L$ containing $C_1$ and $C_2$ constructing maximal chains $A$ in $[0, a]$, $B$ in $[b, 1]$ and gluing them with $C_1$ and $C_2$ respectively. Hence $M_i \cap I = C_i$ and since $|M_i| < \infty$ we have $|M_1| \neq |M_2|$, so (1) implies (2).
I am not really sure about this proof, but it seems right. I'm also not familiar with the notion of supersolvability. It's better to wait for more reasonable answer.
