In this talk, we’ll look at a few examples from Georg Cantor that clarified (mostly through frustration and confusion) several areas of mathematics. There’ll be a little set theory, a little calculus, a little geometry, and a bunch of head scratching.
Given a subalgebra A of a direct product of algebras from some variety V, one can consider Proj, the system of projections of A onto all k-element sets of coordinates. In general, A is not uniquely determined by Proj, but if V happens to have a (k+1)-near unanimity term, then Kirby Baker and Alden Pixley show that this is the case. They also show that if a variety V satisfies this uniqueness property for all subalgebras of direct products of its members, then it must have a (k+1)-ary near unanimity term. In this talk I will consider the following existence question: Given a system of k-fold projections over some direct product of algebras from a variety V, under what circumstances will there exist a subalgebra A of such that Proj? An answer will be given that settles a question posed by George Bergman in a paper from 1977. This is joint work with Libor Barto, Marcin Kozik, and Johnson Tan.