A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic (or torsion) semigroup. But what about the case where there exists $M \in \mathbf N^+$ such that ${\rm ord}_{\mathbb{A}}(a) \le M$ for all $a \in A$? The expression "semigroups of finite exponent" comes to mind, but for some reason it doesn't seem to be so common.
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$\begingroup$ "Periodic of bounded index" is the term I believe. $\endgroup$– Michał MasnyJul 20, 2015 at 10:18
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$\begingroup$ Any (sufficiently strong) evidence in support of your believes? $\endgroup$– Salvo TringaliJul 20, 2015 at 10:31
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$\begingroup$ Bounded torsion is what I would say $\endgroup$– Benjamin SteinbergJul 20, 2015 at 13:34
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$\begingroup$ I was looking forward to your comment, @BenjaminSteinberg, and will probably follow your suggestion, although I'm slightly puzzled by the fact that neither MathSciNet nor Google seem to credit it much, up to few remarkable positive feedbacks (e.g., J. Rhodes has used "Bounded torsion semigroups" in Infinite Iteration of Matrix Semigroups I. Structure Theorem for Torsion Semigroups, J. Algebra 98 (Feb 1986), No. 2, 422–451). $\endgroup$– Salvo TringaliJul 21, 2015 at 18:20
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$\begingroup$ I got this term from Rhodes $\endgroup$– Benjamin SteinbergJul 21, 2015 at 21:32
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