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2
votes
Conjecture about commutative semigroups
As a complement to my previous non-answer, let me show that the OP's conjecture holds for any order-$3$ commutative semigroup $S$ that is not a group (equivalently, that is not a cyclic group of order …
4
votes
Conjecture about commutative semigroups
This is not an answer, but it's too long for a comment: I'm going to show that the OP's conjecture is true in some (admittedly, rather special) cases. It is hoped that this will help to find a counter …