TITLE: Finite lattices whose number of join-irreducibles does not exceed its height
QUESTION [5 upvotes]: In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements.
Briefly, this follows by arranging the subposet $J(L) = \{x_1,\dots,x_n\} \subset L$ so that $j > i \implies x_j \nleq x_i$, and then observing that $0 < x_1 < x_1 \lor x_2 < \dots < x_1 \lor \dots \lor x_n=1$ because join-irreducible elements are join-prime in distributive lattices.


What other natural classes of finite lattices $L$ satisfy $|J(L)| \leq height(L)$?


For example, do join-semidistributive lattices have this property?
Many thanks.

REPLY [4 votes]: Join semi-distributive lattices don't have this property because weak order on $S_n$ is join semi-distributive and doesn't have this property.  (Eg, for $n=3$.)
Lattices satisfying the property you are interested in are called "join-extremal" by George Markowsky, in a paper Primes, irreducibles and extremal lattices. Order 9, 265–290 (1992).  
The Tamari lattices, and their generalization, the Cambrian lattices, all have this property.