Tue, 23 Apr 2024, 1:25 pm MDT
We classify the operations which are minimal above the clone generated by a permutation group G acting on a set. Rosenberg (1986) showed that when G is trivial there are only five types of minimal operations. We show that for most permutation groups there are only three types of minimal operations. More generally, we carry such classification for any permutation group and specify the behaviour of the minimal operations on tuples with two or more elements in the same orbit. Our results are especially helpful for the study of constraint satisfaction problems on omega-categorical structures, where we improve and generalise results of Bodirsky and Chen (2007), and answer some questions of Bodirsky (2021). This is joint work with Michael Pinsker.