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$G$-set; congruence lattice; congruence distributivity; congruence modularity; congruence $n$-permutability
We describe $G$-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine $G$-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence $n$-permutable $G$-sets for $n=2,2.5,3$.
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