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.



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


Many thanks.



Finite lattices whose number of join-irreducibles does not exceed its height
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. 


