Description
The CSP Dichotomy Theorem, independently proved by Bulatov and Zhuk, is a major application of Universal Algebra which brought our field much closer to mainstream mathematics.
Unfortunately, both proofs are quite complicated, and hence further generalizations of the CSP Dichotomy in various possible directions are difficult to achieve.
Our goal is to simplify the proof of the Dichotomy Theorem.
I will report on the progress in this direction made since my previous lecture in Szeged nine months ago and the status of the four open problems I posed in my lecture at the AAA 105 conference, held in Prague this June.
We still focus on the minimal Taylor algebras, as the right setting for any Dichotomy proof.
1. What can't be done: We first provide counterexamples to the attempt to copy-paste the Maroti reduction argument from SMB algebras to minimal Taylor algebras. Moreover, we provide counterexamples to our conjecture from Prague, which was supposed to describe Zhuk centers (in minimal Taylor algebras, those are just the ternary absorbing subuniverses) in terms of Bulatov edges.
2. What has been done: Instead of making the strong as-components act like Zhuk centers, we isolate all the useful properties of sink strong components in the as-graph, and prove that all these properties are satisfied by any minimal center in minimal Taylor algebras.
Thus the right object for understanding ternary absorption is a minimal center, while we already knew from before that the right object for the binary absorption is umax(A).
3. What should be done: The next ingredient in both Dichotomy proofs is a tool for providing "strands", horizontally separated subinstances.
In Zhuk's case that tool is a "bridge", while in Bulatov's case it is the symmetric variant of the inseparability of congruence covers.
We define the two notions, compare them (they are very similar!) and pose the problem of unifying the two.