The paper is a fairly complete and up-to-date survey on two major problems in universal algebra: 1) Is every finite lattice isomorphic to the congruence lattice of a finite algebra? 2) Is every congruence lattice of a finite algebra isomorphic to the congruence lattice of a finite groupoid?
The first problem is unsolved, it is equivalent to a group theoretic question, whether every finite lattice is isomorphic to an interval in the subgroup lattice of a finite group. The proof of this result of the reviewer and P. Pudlák [Algebra Univers. 11, 22-27 (1980; Zbl 0386.06002)], as well as Jónsson’s proof for the characterization of permutational (or minimal) algebras is included in the paper. The second problem was solved in the negative by R. McKenzie [Universal algebra and lattice theory, Lect. Notes Math. 1004, 176-205 (1983; Zbl 0523.06012)], his argument (based on the theory of tame algebras) is sketched here. Since the lattice of subvarieties of a variety is the dual of the lattice of fully invariant congruences of the infinitely generated free algebra of the variety, the survey naturally concludes with W. A. Lampe’s beautiful result giving a necessary condition for the lattice of subvarieties of any variety of algebras.
Anyone who wants to get acquainted with the subject should start with reading this excellent report.
{Note that in Theorem 3.12 the additional remark that \(| G'/H'| =| G/H|\) is not true.}