Abstract. We describe all maximal distributive subsemilattices of a given finite distributive lattice and characterize the covering relation in the poset of finite distributive semilattices. Then we prove that there is no simultaneous representation for the poset of all distributive subsemilattices of the boolean lattice of all subsets of a four-element set by congruence semilattices of finite atomistic lattices.

AMS Subject Classification (1991): 06B10, 06D05

Received November 6, 1991. (Registered under 5532/2009.)