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.
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