Conjugation in abstract semigroups - Borralho, Maria de Fátima Lopes (2020)
On a semigroup S with a xed element c, we can de ne a new binary operation x c y := xcy for all x; y 2 S. Then (S; c) is a semigroup called the variant of S at c. Elements a; b 2 S are said to be primarily conjugate or just p-conjugate, if there exist x; y 2 S1 such that a = xy; b = yx. In groups this coincides with the usual conjugation, but in semigroups, it is not transitive in general. Finding classes of semigroups in which primary conjugacy is transitive is an interesting open problem. Kudryavtseva proved that transitivity holds for completely regular semigroups, and more recently Araújo et al. proved that transitivity also holds in the variants of completely regular semigroups. They did this by introducing a variety W of epigroups containing all completely regular semigroups and their variants, and proved that primary conjugacy is transitive in W. They posed the following problem: is primary conjugacy transitive in the variants of semigroups in W? In this thesis, we answer this a rmatively as part of a more general study of varieties of epigroups and their variants, and we show that for semigroups satisfying ∈ xy {yx, (xy)n} for some n > 1, primary conjugacy is also transitive.