Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$?
Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.
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asked Feb 8, 2022 at 22:17
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4$\begingroup$ I believe they are called duo $\endgroup$– Benjamin SteinbergCommented Feb 8, 2022 at 22:19
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$\begingroup$ Thank you very much for the answer. Indeed there are some papers which use this terminology, but in Wikipedia paper (en.wikipedia.org/wiki/Special_classes_of_semigroups) duo semigroups are absent. $\endgroup$– Taras BanakhCommented Feb 8, 2022 at 22:25
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$\begingroup$ The term comes from ring theory. Wikipedia isn't very complete $\endgroup$– Benjamin SteinbergCommented Feb 8, 2022 at 22:29
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$\begingroup$ The Wikipedia page would use the conjunction of left and right Clifford but to me Clifford means something else. $\endgroup$– Benjamin SteinbergCommented Feb 8, 2022 at 22:33
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$\begingroup$ @BenjaminSteinberg For me also Clifford means something else, namely, to be the union of subgroups. $\endgroup$– Taras BanakhCommented Feb 8, 2022 at 22:47
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1 Answer
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People in factorization theory call a monoid $H$ normalizing if $aH = Ha$ for every $a \in H$; see, e.g.,
A. Geroldinger, Non-commutative Krull monoids: A divisor-theoretic approach and their arithmetic, Osaka J. Math. 50 (2013), 503-539.
In the setting of rings, the term 'normalizing' is used, among others, by Goodearl and Warfield in [An Introduction to Noncommutative Noetherian Rings, LMS Student Texts 61, CUP, 2004], and by Jespers and Okniński in [Noetherian Semigroup Algebras, Algebra Appl. 7, Springer, 2007].
On the other hand, it is sensible to refer to the same objects as duo monoids (as also suggested by Benjamin Steinberg in a comment to the OP), in such a way that a unital ring is duo if and only if its multiplicative monoid is duo (a ring, with or without unity, is said to be duo if every left or right ideal is a two-sided ideal).
Duo rings have been studied (under this name) at least since the late 1950s. To my knowledge, they were first considered by E.H. Feller in
Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958), 79-91.
Since then, the term 'duo' has been regularly used in the literature, including by Thierrin [Canad. Math. Bull., 1960], Brungs [Pacific J. Math., 1975], Courter [Proc. AMS, 1982], Lam in Exercises 22.4A and 22.4B of [A First Course in Noncommutative Rings, GTM 131, Springer, 2001, 2nd ed.], Marks [J. Algebra, 2004], Yu [Glasgow Math. J., 2009], Cossu and T. [J. Algebra, 2023], etc.
answered Feb 20, 2022 at 22:20