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I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in S\implies (\exists a,b\in S)\, xa=yb$$ since then for all $x,y$ there exists $a\in S$ such that $xy=ya.$

Could you give me some concrete examples of noncommutative, cancellative on both sides, normal semigroups (non-groups) or some references to examples?

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  • $\begingroup$ AFAIK, positive braids have this property. $\endgroup$ – Alex Degtyarev Jan 17 '14 at 9:21
  • $\begingroup$ Is there any reason to worry about "normal" cancellative semigroups? Seems this definition is abnormal $\endgroup$ – Victor Jan 19 '14 at 4:57
  • $\begingroup$ @Victor They are inner automorphism invariant in their groups of fractions. $\endgroup$ – Michał Masny Jan 19 '14 at 10:06
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$G\times\{1,2,3,\dots\}$, where $G$ is a nonabelian group.

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positive cone of (nonabelian) totally ordered groups (like the multiplicative group in Hilbert's ordered skew field to show that Pappus theorem is not provable in ordered affine geometry). More examples are given by skew polinomial (and formal power series) rings with respect to a (nonidentity) automorphism of the base (skew) field of coefficients. See also Cohn, free rings and their relations, and the more modern version, Free Ideal Rings and Localization in General Rings (especially section 0.7 and final notes to chapter 0).

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