Tue, 2 Feb 2021, 1 pm MST

Given a subalgebra $A$ of a direct product $\prod_{i \le n} A_i$ of algebras $A_i$ from some variety $V$, one can consider $\mbox{Proj}_k(A)$, the system of projections of $A$ onto all $k$-element sets of coordinates. In general, $A$ is not uniquely determined by $\mbox{Proj}_k(A)$, 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 $\Gamma$ over some direct product $\prod_{i \le n} A_i$ of algebras $A_i$ from a variety $V$, under what circumstances will there exist a subalgebra $A$ of $\prod_{i \le n} A_i$ such that $\Gamma = \mbox{Proj}_k(A)$? 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.