Congruence Lattices of Congruence Semidistributive Algebras

Nearly twenty years ago, two of the authors wrote a paper on congruence lattices of semilattices [9]. The problem of finding a really useful characterization of congruence lattices of finite semilattices seemed too hard for us, so we went on to other things. Thus when Steve Seif asked one of us at the October 1990 meeting of the AMS in Amherst what we had learned in the meantime, the answer was nothing. But Seif’s question prompted us to return to the subject, and we soon found that we had missed at least one nice property: the congruence lattice of a finite semilattice is an upper bounded homomorphic image of a free lattice. This strengthens the well-known fact that congruence lattices of semilattices satisfy the meet semidistributive law SD∧. It turns out that this result admits a striking generalization: if V is a variety of algebras whose congruence lattices are meet semidistributive, then the congruence lattices of finite algebras in V are upper bounded homomorphic images of a free lattice. The proof of this theorem takes us into the realm of tame congruence theory, and with modest additional effort we are able to find strong restrictions on the structure of the lattice Lv(W) of subvarieties of an arbitrary locally finite variety W. Stimulated by Viktor Gorbunov’s talk at the Jonsson Symposium in Iceland in July 1990, and the corresponding draft of [12] which he provided us, we went on to ask if this type of result might apply to lattices of quasivarieties. It is known that the lattice Lq(K) of all quasivarieties contained in a quasivariety K satisfies SD∨ [11], and the improved (finite) version states that if K is a locally finite quasivariety of finite type and Lq(K) is finite, then it is a lower bounded homomorphic image of a free lattice. There are natural generalizations of this theorem for varieties which are not locally finite. Perhaps an analogy with the modular and Arguesian laws provides a good way to interpret these results. Dedekind devised the modular law to capture the permutability of normal subgroup lattices, but the Arguesian law is now recognized to be a more accurate reflection of this property. Similarly, congruence lattices of semilattices satisfy SD∧, but (at least in the finite case) upper boundedness is a stronger property which provides a better description of their structure. As the Arguesian law is not sufficient to characterize normal subgroup lattices, neither does lower boundedness characterize congruence lattices of finite semilattices. Nonetheless, the Arguesian law and upper boundedness, respectively, play a significant role in refining our understanding of these classes of lattices. Since completing the draft of this paper, we learned of significant progress on the