3. Modularity for lattices of finite length

Theorem. The following conditions on a lattice $L$ of finite length are equivalent to conditions (1)-(5) above.

(6) In $L$, $a$ and $b$ cover $a \wedge b$ $\Leftrightarrow$ $a \vee b$ covers $a$ and $b$.

(7) The height function $h$ in $L$ obeys $h(a \vee b) + h(a \wedge b) = h(a) + h(b)$.

Theorem. Any modular lattice of finite length obeys the Jordan-Dedekind chain condition:

In any interval $[a,b]$, any two maximal chains have the same length.

Remarks.

(i) This last theorem is actually true for any ``semimodular'' lattice of finite length. A lattice of finite length is semimodular if it obeys `` $\Rightarrow$'' of condition (4).

(ii) The condition (J-D) shows that the height function $h$ is a well-behaved rank function, in that for $a \leq b$, $b$ covers $a$ if and only if $h(b) = h(a) + 1$.