Results and problems on congruence lattice representations

This is the writeup of a talk given at the conference in honor of Walter Taylor. The key to congruence representations is Mal'cev's lemma describing the congruence generated by a pair of elements. It says that this congruence is determined by the unary polynomials of the algebra, or by what Gratzer calls unary algebraic functions. Congruence representation constructions provide a set of equivalence relations with the desired order structure and a set of unary polynomials doing what Mal'cev's lemma says needs to be done. For an infinite algebra, the number of compact congruences is no more than the number of elements. Any algebra with at least one at least binary operation seems to have at least as many unary polynomials as elements. So at first one might expect that one could represent any algebraic lattice as the congruence lattice of a groupoid (an algebra having just one operation which is binary), but that is not the case, as we shall see.