Dmitriy Zhuk (Charles University, Prague, Czechia), Clones on 3 Elements: A New Hope

Tue, 14 Mar 2023, 1:25 pm MDT

In 2017 Moiseev found 2 079 040 clones on 3 elements definable by binary relations using a computer. It is clear that only computer can deal with so many clones but another result from 2019 showed that even computers have their limitations. Moore proved that they cannot check whether a clone given by generating operations is finitely related (definable by a relation). These results made us believe that even a computer description of all the clones on 3 elements would never appear.
However, not all clones are essentially different, and one might try to characterise clones modulo clone homomorphisms or minor preserving maps (only h1-identities are preserved). In 2022 we showed that there are only countably many clones of self-dual operations on 3-elements modulo a minor preserving map. Then using a computer we collapsed 2 079 040 into a very small number and now we hope that a complete characterization of all clones on 3 elements modulo minor preserving maps is possible.
Joint work with Libor Barto, Jan Adam Zahálka, Albert Vucaj, and Florian Starke.